Misaki and Ayane aren't co-stars. They aren't actresses. They're an idol unit. Why does that matter? Because unlike actresses, for idols even their friendships are part of their job. The job of an idol isn't just to perform on stage, it's to create a "character" to appeal to the fans. The idol has to show that "character" all the time. They are meant to represent a fantasy for the fans, after all.
So Misaki can't realistically stops being close to Ayane. Her relationship with Ayane is part of the idol package they're selling to the fans.
Yes, that's the professional face they have to present. That doesn't mean they have to be as close in private life as they portray for the fans, though. I mean, isn't she supposed to be "friends" with the other girls in the group? Maybe I'm misunderstanding something, but seems to me like normally their behind-the-scenes relationship is more professional.
Nagisa should have known what she was getting into from the get go. The Japanese are much more aware of what it means to be an idol than we are.
Nagisa sees the public stuff and doesn't mind. However, when Misaki herself sends a picture of her and Ayane being close, that is when it gets a bit much.
There is a reason why the "you vs the [girl/guy] [he/she] tells you not to worry about" meme exists.
I don't think all Nagisa wanted was understanding. I think she wanted a solution, and that's the problem because there was no solution, short of Misaki quitting her job.
All she needed was a reaffirmation of their love. It was a cry for attention. The reason Misaki flew off the handle was because she had feelings for Ayane. She could've just said that her feelings for Nagisa are different from her feelings for Ayane. She could maybe even have said that she'd talk to the other girls/her manager to see if there's anything they can do. There's dozens of ways to handle a bit of jealousy that don't involve just dropping everything right here, right now.