Couldn't the recruits simply take the money and still go wherever they wanted?
First, the government is involved in every transaction between people because that's what society actually wants. Why do you think contracts and agreements have any recourse to be enforced? It'd be anarchy if the government didn't set the laws and rules by which people could transact. Second, it's contradictory for you to want the NCAA to intervene in private transactions and not the government. Plus the NCAA is run by universities, many of which are government institutions.
Most universities are not private companies - that's a very big difference. And no one is saying that Twitter and Facebook are exempt from laws about paying their employees, which is what the NCAA is doing.