The NCAA has been sued before. It can afford lawsuits as it is funded by its member institutions. It cannot afford to lose member institutions, largely for that very reason.
You merge multiple topics into one. If the "student athlete" went away, there would be no current NCAA. Its existence is based on that status. In fact it created the term "student athlete". The NCAA, speaking for its member institutions, wants that status to remain. So, of course, the member institutions don't want student athletes to be professionals as that changes the entire model and resulting bottom lines.
The NCAA is the "governing" body of 347 DI schools who voluntarily join. The billions you refer to do not go to the NCAA (meaning stay there), they largely go through the NCAA back to the member institutions and conferences. The NCAA receives money, then passes it back to their member institutions and activities. See
http://www.ncaa.org/about/where-does-money-go.
That revenue has nothing to do with things like conference TV and media packages. Those are a big part of the "billions" people like to throw around, but those don't belong to the NCAA. Those go to schools and conferences. However, at least now, if you aren't an NCAA member, you won't be in a power 5 conference. If you aren't in a power 5 conference, you won't get to share in the big TV revenues and won't be able to participate in things like bowl games or other NCAA championship events. So you need to be an NCAA member, even though it is voluntary.
And the NCAA needs its member institutions.