This is part of the problem - I feel that we don't push her enough. My DM is controlling and I try to be the opposite of how she was.
Is this more to do with your DM, and wanting to be as far away from her style of parenting as you can get - but then wondering if that's right for your daughter?
There's nothing that you've said that's worrying, really. She's happy, fine at school, does have friends there, will go to parties/cinema etc sometimes. She's just not overly sociable outside of school, and you feel that she lacks hobbies. She'd probably say that TikTok is a hobby, I have friends who definitely would consider it one of theirs!
The crying over a friend is a bit of a non-event. It sounds like a one-off, and she could well have been pretty upset that you invited someone over that she then had to play with, and she didn't want to. I don't particularly enjoy having to entertain people that I don't get on with.
I am quite sociable, but plenty of people aren't - or are, but don't meet anyone that they particularly want to spend excess time with at school. She's able to make friends at school, to keep them, to communicate and socialise....
So really you'd be taking her to a psychologist because you're not overly happy with who she seems to be, and that would be tough to explain, potentially now but almost definitely in the future when she's older and realises that it's not normal.
Do you still get any psychological help? Could you talk about this with them, to see if it's actually an issue with you not feeling that you are parenting right because of your DM, rather than anything to do with your daughter?