I don't think you come across as being an a$$ but there does seem to be conflict between you and your gf about what your role with her children is.

You're not married and not living together, and these are not your children. If she were just a friend of yours, not a girlfriend, would you give her your unsought opinion on her parenting decisions? Ask to take her child to an event and then argue with her if she says No?

You've been dating 2+ years, and are not married. Why is that? Your gf needs to protect her children from "serial dads". If things don't work out with you 2, and she moves on, her children as well as she, will have to let you go. It makes sense to me that she would limit your role in her/their life.

It sounds like you and she need to discuss your quasi-parent role. It's very nice you want to be a good role model for her kids, but your uncommitted relationship with their mother is going to limit what your role should be.

As far as the silent treatment and the leave-me-alone-for-the-weekend, I don't know, that's a separate thing. Sometimes people just need some space to think. Sometimes they have an unhealthy running-away trick they do. I'd leave her alone for the weekend then call on Monday. Maybe set up a child-free time to talk? Good luck!