Universities do exist to add value to society through education and dissemination of information. Seems like a pretty clear charter if you ask me.

Don't some universities exist to make money?

Universities are generally non-profit. For profit ones exist but are generally regarded as scammy.

Of course, one could argue that universities exist to grant generous salaries to highly ranked administrators, who are mostly politicians.

It doesn't matter 'why' something exists, what matters is what those things that do exist are actually useful for.

Screw drivers exist to turn screws, but you are perfectly justified to use them for whatever you need should you find one in your hand.