If government were not as influential as you claim, there is absolutely no reason the wealthy would use the resources they buy power with to provide goods and services, especially to the poor and needy who cannot afford them as is. In fact, you completely ignore the robber barons and a time when less regulation DID exist, and society was even less free and equal.
There is NO such thing as a truly free market. Free markets are an ideal simplification, like calculations done in a frictionless vacuum. The role of government in the economy is to regulate in such a way that a sort of free market is created. That competition exists and everyone benefits.
Unfortunately that power is not always used correctly. Regulation is difficult, oftentimes it doesn't work as expected. But that does not mean the problem is regulation itself. The problem is specifically the regulations that exist today. That makes sense, when you realize that half the government is trying to prove that the government is terrible. I really wouldn't trust them with regulating anything.
The greatest time in recent American history was created by socialistic, big government programs and regulations. Period. The massive amounts of wealth generated during that time went to every American, not just the few, because of those policies. Removing government is not the answer, we need to fix it.