Kakashi's talk with Sakura was, ironically, Kakashi's best foot forward in reaching Sasuke. Unfortunately, after Kakashi's apology to Sakura, his interaction with Sasuke is all downhill from there and his attempts to reach Sasuke go sour afterwards. But he had a great start with the apology to Sakura at least. Kakashi came clean to his pupil and admitted his faults. It's not just internally berating himself for his faults, but an open admittance to Sakura that he'd behaved irresponsibly and failed as a teacher. It's a nice moment of Kakashi confronting and admitting his mistakes, apologizing for being responsible for Team 7's current sad state.

It's nice that Kakashi acknowledges how he'd hurt Sakura with his lies, because finally Kakashi is treating Sakura like a person, not just some idiotic child he can keep tricking with lies and hollow reassurances.

Plus Kakashi is finally being self-aware. Team 7's downfall was partially due to Kakashi's irresponsible and immature leadership, so at least Kakashi can finally admit it and take the blame. Kakashi admits his faults and failures: he is a terrible teacher.

Considering Sasuke's current frame of mind and how upset and angry he is, having just fought Danzo and nearly been assassinated by his former teammate and insulted by his former teacher for having fallen so far – Sasuke responds unexpectedly positively towards Kakashi's apology.

Kakashi's apology diffuses some of Sasuke's aggression for the moment. Sasuke seemed initially tense with battle-readiness, expecting Team 7 to be on a mission to kill him. Though Sakura killing Sasuke wasn't actually a team mission, since Kakashi didn't know about Sakura's plan. Which saves Team 7 face a bit, since it's not some village mission they are carrying out, but rather a personal moral obligation for Sakura. Additionally, Kakashi's phrasing is neutral enough that Sasuke might infer that it hasn't been decided yet if Kakashi intends to kill Sasuke or if Kakashi is going to try to talk Sasuke down instead. So when Kakashi clarifies that Sakura is there to kill Sasuke, Sasuke can deduce that Sakura is acting on her own, meaning that Team 7 isn't on an official mission to kill him after all, and Sasuke curiously awaits to see what Kakashi's next move will be.

Kakashi isn't talking to Sasuke of course, but Sasuke listens to Kakashi's exchange with Sakura anyway, watching attentively as Kakashi apologized to Sakura for being the reason that the team was torn apart. Despite his earlier aggressive smirk at Kakashi's arrival, Sasuke sobers and his expression after Kakashi's apology to Sakura is calm. He doesn't look aggressive or angry or upset. He rather looks thoughtful, tired, and weary. His expression has softened, meaning that Kakashi's words up to this point are having a positive effect, or at the very least, not incensing Sasuke further. 

Just for kicks, I posted this panel of Sasuke's expression without context and asked fans how they'd describe Sasuke's expression. Here are the replies I got:

"Curious"

"Sad, hurt, betrayed, worried, feels alone..."

"He looks like he's trying to consider something or pondering a new idea (having a realization). It's difficult to describe."

"I felt like he's feeling empty."

"Uncaring."

"Exhausted."

"Tired."

"He seems to be lost in thought. Despite the fact that black is his usual eye color, Sasuke's eyes seem even darker than usual. He looks like life has robbed him off some years, like he's older than he actually is. He looks sad and nostalgic, as if wishing, deep down, to go back to the more innocent times of his life; his childhood, but also, to an extent, his time on team 7."

"Mentally agreeing with something he just heard."

"I would say he looks like he is trying to conceal his emotions, but the "..." might show otherwise. Of course he also looks exhausted from the fight against Danzo etc."

Notably angry, hostile, or aggressive were none of the descriptors. Kakashi's apology caught Sasuke's attention and interest, so now Sasuke observes closely, waiting to see what Kakashi will say next. What Sasuke wants to know is: What now? What next? If Kakashi admits he messed up, what will Kakashi do to fix his mistakes?

Sakura is kind enough to protest that Kakashi isn't a bad teacher, but Sasuke seems in full agreement with Kakashi's statements: Kakashi is a terrible teacher, and Kakashi contributed to driving Team 7 apart.

Kakashi's confession that his carelessness split the team apart is evidently a confession Sasuke finds interesting. Sasuke is intrigued by his ex-sensei admitting that although Sasuke's desertion was messed up, it was in part the irresponsibility and carelessness of the adults around him that drove Sasuke to such actions.

Furthermore, Kakashi admits that he once said something irresponsible to Sakura by trying to reassure her that everything would be fine, that may he was just trying to cover up his own anxiety and ignore the problem right in front of his eyes because he was too lazy and careless to deal with such a difficult problem. Blinding himself to the truth and trying to convince himself with lies was easier and more convenient.

Sasuke's reaction isn't angry, or mocking, or jeering. Rather, Kakashi has evoked the "thoughtful ellipse" response in Sasuke, a thought bubble always indicating a Naruto character is thinking seriously and thoughtfully as they process the events transpiring around them, chewing over them and reacting to them in a meaningful and significant way.

All these things Kakashi said catch Sasuke's rapt attention, so when Kakashi ends his dialogue with Sakura and turns his attention to Sasuke, opening with a comment about how he doesn't like to repeat himself, Kakashi has earned Sasuke's full attention. Sasuke leans forward slightly, focused fully on Kakashi, diligently anticipating what Kakashi will say next.

Sasuke's reaction makes me think that Kakashi's apology to Sakura is exactly what Sasuke wants to hear from Kakashi: an apology.

Sasuke's reaction to Kakashi's apology here coupled with Sasuke later blaming Itachi for turning him into a criminal makes it appear that Sasuke is desperately awaiting an adult to come help and rectify the situation. Adult are supposed to protect children (both physically and mentally), establish order, administer care and comfort and support, personally handle the most difficult challenges, offer guidance, and right wrongs.

A huge problem with Sasuke's psyche and the causation between his fall to the dark side is steeped in the fact that none of the adults in Sasuke's life are acting like adults. Kakashi sure isn't. And Sasuke spending his life surrounding by immature, irresponsible, uncaring adults has resulted in serious behavioral issues.

It's also led him to a strange frame of mind in how he reacts and responds to adults; Sasuke is caught in a weird state of wanting total independence and freedom from others while simultaneously experiencing a desperate, innate longing for adults to behave like adults and establish law, order, and justice and administer care to children, as adults are supposed to do. Sasuke is torn between desperately wanting to trust and rely on adults while also knowing full well that is logically something he can never ever do, firstly because he has severe trust issues, and secondly because the adults in his world all are completely unreliable, undependable, and untrustworthy.

This leads to some odd nuances in Sasuke's interactions with Kakashi, Orochimaru, and Obito especially, all three being the primary adult influences in his life during his post-massacre years, and all three who majorly let him down. Sasuke wanted to be able to rely on and trust an adult, but doing so would be incredibly foolish and only lead to them disappointing him and letting him down even more, not to mention possibly getting him killed. They weren't dependable, nor were they admirable or trustworthy mentors/guides.

At this point, I'd even say it's stunted Sasuke's journey to psychological adulthood in a weird way, since he's still caught in a strange state of wanting childlike dependence on adults, but because no adults have ever filled that void (since he was 7 years old), it's stunted his psychological growth toward mental maturity and independence a bit.

What Sasuke wants is for Kakashi to apologize; same with Itachi, and maybe virtually every other adult in existence that had any significant bearing on Sasuke's life (Fugaku and Mikoto are the exceptions, of course, but they actually took care of their son why there were alive, so Sasuke could depend on those adults and they didn't leave Sasuke with a deficit of care the way the rest of the prominent adults in Sasuke's life after the massacre did).

Perhaps it's presumptuous, arrogant, and self-victimizing for Sasuke to want an apology from Kakashi, especially considering how much Sasuke personally has wronged Kakashi. However, I don't think Sasuke exactly has a problem with admitting his own flaws and faults (considering he's already admitted to Tobi that his behavior may seem childish to others, he just doesn't care, and in the war arc he later himself admits he needs to grow up and stop acting like a child).

But it seems that Sasuke is searching for is affirmation that he isn't the only one to blame for the person he turned out to be. He wants those who failed him to admit and acknowledge how much they let him down, and that even though he is ultimately at fault for his decisions and the choices he made of his own volition and free will, their failures played a part in him making the choices he did.

Kakashi's observation that Sasuke is "a product of circumstances" is good and all, and it's nice to see Kakashi be observant about the flaws of his world that cause problems in people like Sasuke later down the road. At the same time, Kakashi is detaching himself from the mess that is Sasuke, which strikes me as shirking responsibility and denying the consequences of his own actions, since Kakashi doesn't wonder if there was ever anything he himself could have done differently to help Sasuke.

Generally if one is going to acknowledge something being a result of a societal problem, that demands self-reflection, because Kakashi is a part of the world too and thus part of society, and thus potentially a part of the problem, especially since in this case since Kakashi was one of Sasuke's primary influences during Sasuke's time as a ninja. But we see no such introspection from Kakashi. Instead of learning from the tragedy that is Sasuke and wondering how he can improve himself and not let down others like him down in the future or even how he can help Sasuke now, Kakashi distances himself from responsibility, acting like he never had anything to do with Sasuke, and blames instead a vague and distance "them" and "society" and "our times" (as if Kakashi isn't a part of society or a part of the current era, even though he is), because that's easier and more convenient than actually holding himself fully accountable and thinking that, had he been more diligent, caring, and responsible, maybe Sasuke might not be the mess he is right now.

Anyway, Kakashi confessing his mistakes sets a good precedent. After all, how could Kakashi expect Sasuke to admit his misdeeds and faults and learn from them if Kakashi is not willing to do the same? By confessing his failures and how his mistakes caused his team a lot of hurt, Kakashi addresses both his mistakes and the harm they caused others. It's a great way to set an example for Sasuke. Kakashi isn't merely vaguely lecturing Sasuke to "do better", he is personally modeling repentance and acknowledging one's crimes by admitting his faults himself.

Actions speak louder than words, and by admitting his faults, Kakashi's actions speak loudly to Sasuke, which is why Sasuke watches with a surprisingly calm and thoughtful expression.

But then Kakashi wrecks everything by reverting back to the same "give up revenge" lecture again.

If Kakashi isn't willing to change his ways, then Sasuke isn't either. And Kakashi's promising opening is ruined.

Had Kakashi not wrecked the situation by reverting back to the same "revenge is bad" spiel, this run-in might have turned out quite differently. At the very least, it would've been interesting to see what direction it would have gone. Of course, I'm not expecting anything as amazing as Sasuke agreeing to stop his revenge. I'm not even expecting Sasuke to not end up attempting to kill Kakashi by the end. But Kakashi could have potentially done a lot to cause Sasuke to at least waver in his course, particularly had Kakashi tried to discuss recent events.

In this situation, probably the best thing for Kakashi to do would be to extend his apology to Sasuke. If Kakashi truly feels guilty about tearing the team apart, then Sasuke deserves an apology as much as Sakura. After all, Sasuke is another victim of Kakashi's irresponsible teaching. If Kakashi wants to express goodwill towards Sasuke and attempt negotiations that will calm Sasuke down, an apology and admittance of fault is a great place to start. Since Kakashi already apologized to Sakura, he's not above being able to admit his faults, so why doesn't he apologize to Sasuke too for being a bad teacher? Why not start with apologies rather than immediately jumping into a lecture?

By immediately reverting to lecturing Sasuke in an impersonal, detached way, Kakashi destroys all the credibility he'd just tenuously established with Sasuke. From his apology to Sakura, Kakashi had shown that he was able to admit his mistakes, that he possessed the capacity to learn from them and use his failures to improve himself.