The operative word there is "might." Greece has free education. Their educated youth are fleeing as fast as they can. Not very profitable. All that free education also somehow failed to prevent their chronic debt crisis and general decline. It's almost as if economic prosperity is more involved than just distributing free stuff........
Except the actual cost of college is not free, and for a lot of material covered in colleges it is an extremely inefficient way to convey that information, in both time and monetary cost.
A large part of the problem is that, in a glut of labor, employers use degrees as litmus tests of commitment rather than ability. If you can just ignore anyone who doesn't have a bachelors, you will, because why not? But that doesn't mean the bachelors is, at the macroeconomic level, worth it to society to push everyone to get.