First of all, this post contains spoilers, so if you didn't complete KS, just skip over this post.
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There were a wide variety of friendships in KS and that was pretty admirable. Many high-school VNs have casts that are either all isolated from each other or all friends. (ideal for harem endings) The various friendships add a lot of depth to each of the girls because it shows how they interact with others, not just with the protagonist. That said, I disagree with some of your statements.
For example, Rin's and Emi's friendship was depressing; they could not even be considered friends. I liked how this subject was handled, considering the girls' personalities.
I agree, there's a depressing aspect to Emi's and Rin's friendship, but frankly there are depressing aspects about all the three friendships among the female cast. I think rather than friends, Emi and Rin could be considered buddies. They live next to each other, Emi likes to use Rin to keep in touch with her nurturing side and for Rin (who has trouble with shirts) that's pretty convenient. They like to hang out together, mostly because it suits their own preferences. Emi has severe trouble letting people close, so the distant Rin (who lives in her own world most of the time) feels safe as someone to hang out with. Emi can chatter to her heart's content and feel secure in the knowledge that Rin's too preoccupied with clouds and the color of the paint on nearby woodwork to come up with personal questions that make her uncomfortable. Rin admitted that Emi's empty chatter is useful for clearing her head and Emi herself tends to keep Rin from getting into too much trouble. (she sees to it that Rin eats regularly and attends class) Emi also feels safe to Rin because Emi is content to just hang out with her without trying to make sense of her. Emi and Rin are two people who can't get close to others and are friends out of convenience. Neither has a particularly strong emotional connection to the other, meaning that even at the good end of Emi's or Rin's path, they're unlikely to get closer. In fact, chances are that if one of the two were to try and bridge the gap, their friendship would quickly dissipate. And yeah, that's rather tragic.
Even though I did never really like Shizune and, instead, really liked Misha, I think their friendship was one of the best throughout the game. Even after rejecting Misha, Shizune did not let them grow apart, and they still managed to be friends. After something like that, I believe that it would be pretty hard to maintain their closeness, yet they did.
Shizune not letting Misha "grow apart" from her was, IMHO, a screw-up on Shizzy's part and one that might have destroyed their friendship in the long run if it hadn't been for Hisao's mediating. Shizune was more than a friend to Misha and also more than a crush. She was Misha's "responsibility". Shizune likes to play a very high-profile role in whatever she participates in...student council, class rep duties, everything... In order to function on that level, she kinda needs Misha to be around...Shizune HATES communicating with written notes. Shizune knew that and so did Misha, so when Shizune rejected her advances, Misha stuck around Shizune 24/7 like she did before. And Shizune let her...part out of convenience and partly due to Shizune misguidedly assuming Misha shared her compartualized way of thinking. In practice, being romantically rejected by someone and still sticking to them like glue afterwards is the WORST way to get over it. It made the friendship agonizing for Misha and prevented her from moving on properly. If Shizune had rejected Misha and then let her have some space, encouraging her to make some new friends and telling her that she'd welcome her back as a friend after she sorted out her feelings, their friendship would have been a lot more healthy than it was at the start of KS even if it meant Shizune'd be without an interpretor for a while. In the long run, Shizune's and Misha's friendship will remain a bit of a balance act since Misha's feelings might return if the two draw TOO close, but there's enough non-sexual affection between them to remain pretty close with one another.
Another kind of friendship I noticed was between Hanako and Akira. Before the game took place, I suspect that akira was more of a friend than Lilly was; we all know what Hanako thought about Lilly at her bad ending. Hell, Akira was such a nice person that she could easily befriend everyone she knew; and I loved the relationship between Akira and Lilly, it was a really heartwarming one.
Akira is a very likable person though also tiring at times. The strong emotional bond between her and her sister is heartwarming though. The fact she took care of Lilly after their parents left Japan instead of leaving Lilly with their grandparents shows that despite her willingness to let a bunch of teenagers get drunk of their mind, Akira is a very responsible person at heart. Akira's relationship with Hanako is best described as casual friendship. Hanako is comfortable around her, but I doubt she'd be very likely to confide things in Akira that she wouldn't confide in Lilly. Akira's also unlikely to just drop by at Yamaku purely for Hanako and I doubt the two would be very likely to keep in touch if Lilly moved back to Scotland. But I think Akira's casual friendship was still very nice for Hanako to have.
An interesting thing to look at is the relationship between Akira and Lilly and the relationship between Lilly and Hanako. It's essentially the same relationship in different stages. Akira started out as the older sister who's also a parental figure to Lilly who took care of her after she lost her parents. But now Lilly has grown up emotionally and the two hang out and banter as equals. Lilly is also the older sister who's also a parental figure to Hanako who took care of her after she lost her parents. The close sisterly relationship Lilly and Akira share is also the exact kind of bond Hanako desires to have with Lilly.
I think you're doing Lilly's and Hanako's friendship a strong disservice by using only Hanako's bad end as a guideline to judge it by. While it's true Hanako shouted she hated Lilly during her bad ending, Hanako was in the middle of a massive depression and she felt hurt because Hisao suggested he and Lilly were worrying about her behind her back. Hanako's true feelings for Lilly are far more nuanced. She dislikes Lilly's worrying, but part of this is her own damaged self-esteem. Lilly herself reveals in her route that she didn't befriend Hanako out of pity. Even though Lilly on her own is unable to get Hanako to get out of her shell, the comfort and stability she provides is nevertheless something very important to Hanako. This comfort and stability is why Hanako initially sought out Lilly to begin with. Despite the sometimes ambigious nature of their friendship (friends or mother/daughter?), Hanako's and Lilly's friendship is undoubtedly the most profound one of all the girls. Shizune and Misha might get close, but ultimately never as close as Misha would like. For Hanako and Lilly though, the idea of an ideal friendship...a strong sister-like bond with the other, is exactly the same for both. It's mostly Hanako's self-esteem issues and Lilly's tendancy not to confide her own problems in others that's holding them back, but both Lilly's and Hanako's good end can plausibly lead to the two resolving that. Hanako and Lilly are the only pairing who both openly admit they genuinely love the other...something not seen in any other friendship in KS.