Ow boy, I love these kinds of posts. Lots to unpack here. Let's get to it.
They need her. Scott has just a handful of X-Men to try and push back the tide of yet another mutant extinction event horizon. It would be foolish to refuse the assistance of someone who's been in this position many times, knows the X-Men's personnel, tactics and methodology and is someone he personally trusts.
The current status quo in Uncanny is just as dire for Emma as it is for the X-Men. She's a highly visible, public mutant who also happens to be very powerful. If she's not hiding under the biggest rock she can find right now, she's asking for imprisonment or death. She has everything to gain and nothing to lose in joining forces with the X-Men. It's not like she can fly under the radar. She's been on national TV right beside Scott several times, and she's got to be on the radar of every anti-mutant organization and death squad on the planet. The difference is that she's not surrounded by people she knows will have her back. A few payoffs or political leverage against the right people and she's compromised.
I can't speak for "people" but for myself I can say I enjoy reading her. She's unique among the X-Men in many ways. There's very few characters that can match her will and drive while still being multifaceted. She injects some humor into the books and generally keeps the X-Men from living entirely in their ivory tower. That's a good thing, and no other character really fills that role.
Emma certainly cares about some of the X-Men. Her love isn't universal certainly. She tends to be much harder on people who's very attitudes put the X-Men at risk. But for those with the intelligence to see Emma's value, she has been shown to care for them. I really do think she has the X-Men's best interests at heart. It's just not the kind and gentle, kind of puppy affection we see from Storm or Jean. Just because someone pushes your buttons and drives you to see the world more realistically, doesn't mean they don't care about you. Emma's certainly no angel, but in the last 15 years of publication, she's been a lot better person than many give her credit for.
There were many times that it would have been in Emma's best interests to cut and run. Definitely during the Utopia era, she had very few reasons to stay but even fewer once the team was isolated at the New Xavier School, pushing an agenda that was likely to get them in jail or dead. Emma stuck through all those years, with a fierce determination to see mutants survive. That's not the actions of someone who doesn't care. She's has been shown to care for her students and charges explicitly.
As for Emma's sharp tongue, I think people tend to forget that Emma's mostly just giving as good as she gets. Go back and read Astonishing and tell me that Kitty isn't continuously picking a fight with her. This is something Emma deals with consistently when it comes to the X-Men. A number of them treat her like garbage. She sticks around because it's where she can do the most good. She knows it, and at the end of the day, they know it too. They just hate to admit it.