Okay yeah, you're right, a story focused on her as the FMC would also be good, and while I wouldn't have rooted for her if she had been introduced at a point where she felt like an actual choice for the girl Ayumu ends up with, she would totally have been a worthy love rival, like one that actually might be a threat. To be fair, I would have despised her if that was the case because I don't like love rivals, but at least she wouldn't be such a sad character 😅I do get why people like Rin; in isolation she's a great character, and I'd have happily read a series with her as the FL. Or heck, even if she had come in much earlier and it wasn't 100% clear that Ayumi had already made his choice I might have been tempted to cheer for her.
But as you said, it just feels kind of sad watching her still fight a hopeless battle. It would have been much more satisfying to see her make her move, get rejected, and then hopefully find a new partner. Then it would feel fine to have her still tease and encourage Ayumi, because it would feel more like she was helping a friend she cared about instead of trying to break up a relationship.
And yeah, she could have been a great addition to this story if we had seen her move on, her teasing for example could have been a great driving force for Urushi, because while Rin, Ayumu and we as the audience would see it as teasing between friends, she wouldn't and therefore have a reason to tie him down faster. Also as you said her helping him as a friend would feel a lot better if she wasn't also still gunning for him.