Well obviously Mikage Kotooka has some background stuff going on, I'm thinking either she has a crush on Tsukasa Shiratori, knows about Shiratori's feelings towards Washio, or she had an incredibly bad experience with love in the past (or some combination of the three). Nadeshiko Washio probably thinks she is helping Shiratori out and repaying her for the emotional support.
Wait, is it not already 100% clear that Kotooka is in love with Shiratori? I thought it had been confirmed already. Might be wrong about that, but that's how I'm expecting it to go, at least - Kotooka fell for Shiratori, gave up on love because she sees it's unrequited or is sceptical of same-sex relationships, and has been dating boys indiscriminately because she doesn't trust her own ability to understand what love is and/or hopes that random boys will be a good distraction. She's now pushing that hopelessness onto Shiratori and is encouraging her to date a random boy too, because she wants to make it clear to herself that her feelings are impossible.
She has a reason to be pushy, even if it's a twisted reason that's going to make everyone miserable. Washio just seems like she's trying to be helpful without ever bothering to ask what the person she's "helping" wants or needs, prioritising some generic ideal of what girls "should" want out of life over whether or not Shiratori feels comfortable with it, or in the worst-case-scenario, prioritising some random boy's feelings of mostly imaginary, slightly stalkery love over the girl who's ostensibly one of her best friends. There's also the possibility that Washio suspects Kotooka's feelings and is trying to remove Shiratori as a rival, but she doesn't seem like that sort of person?
(I'm still leaving open the possibility that Glasses Boy saw Shiratori's brother while he was crossdressing and fell in love with HIM, and maybe they had a conversation that the brother forgot about, and when he tried to track him down he mistook one Shiratori for the other. That would make his sudden confession a lot more understandable and less creepy, but as far as any of the girls are aware, this is the first time they've ever met.)
Really, Shiratori also should take some of the blame for not making it clear that she has someone she is currently crushing on. I mean the guy is essentially prince charming and if you say that you currently have no one you like, there really isn't any logical reason why anyone would say you shouldn't at least meet him.
I mean, no? She shouldn't actually need a reason not to meet someone beyond "I don't feel like it", especially when that person followed her to her school to confess, apparently without ever having a single conversation with her, which I don't really think of as ideal Prince Charming behaviour.
She's shown that she's fairly reluctant about doing anything with him already, if they cared enough to notice, but they've still been pushing her into "giving him a chance" and arranging dates for her anyway. The fact that she happens to be in love with someone already is kind of immaterial, at this point. If those are the magic words that would make them back off when "I don't know, I thought he was a bit weird, I'm not really interested" is arbitrarily deemed illogical and insufficient, then that makes them bad friends. If we're being generous, we can say that they have their own sympathetic and understandable reasons for being bad friends, but that doesn't really change the fact that they're being bad friends.