Are private institutions allowed to take away things like freedom of speech and assembly just like that?
What do you mean by "take away" those things? Not allow their property to be used for the dissemination of viewpoints they find objectionable? Of course.
I say "take away" because prior to this decision, the school's Democratic club was able to meet on school ground and hold their meetings. Then they said that such meetings were banned on campus. I think it's wrong that they're able to take away such basic things like the freedom to assembly, private school or not.