We've discussed this before, actually.
Jazz wanted to a different college, but mama said no. She can say that's because she wanted Jazz to be close to her to protect her and help her, but as you said, he could have attended a local U. The reason Jeanette wanted Jazz to go to Harvard is because he's the child she's living through. Her precious "girl" is not going to some cheap ass college in Florida like the other three, right? No, they have invested so much on Jazz, so he goes to Harvard and become a lawy--- an astrologist or something.
But now, she needs Jazz to be close to keep filming the show? Pffff.