Is there a way to have low cost health care and college that would be afforable for all without the government?

The government seems to be the answer for all of societies ills. Despite that the government is extremely inefficient and most things would be ruined or heavily impacted by corruption. I believe we could have free market insurance and be allowed to form large groups thru credit unions or other organizations. Vouchers could be given from the government for people in poverty. We could also form large groups for college and have the billionaires who complain of low taxes chip in. We could also get the price of college back down to how it was in the late 80’s by forcing colleges to compete on prices.