Hoover hired Treasury Secretary Andrew Mellon, who believed in the "leave it alone" approach. Hoover may have had the sense not to follow that advice, but hiring Mellon indicates Hoover's underlying philosophy.
Nonsense. What Hoover actually did while on office indicates his "underlying philosophy", which was far from "leave it alone". In any case, regardless of any philosophy, the fact is that he intervened to an unprecedented extent, with predictable results.
You said that Hoover "let the free market work on its own". Hiring someone who believed in that approach and then proceeding to ignore their advice does not constitute letting the market work on its own.
There are varying accounts regarding the Panic of 1837. Quoting Wikipedia:
Most economists also agree that there was a brief recovery from 1838 to 1839, which then ended as the Bank of England and Dutch creditors raised interest rates. However, economic historian Peter Temin has argued that, when corrected for deflation, the economy actually grew after 1838. According to economist and historian Murray Rothbard, between 1839 and 1843, real consumption increased by 21 percent and real gross national product increased by 16 percent, despite the fact that real investment fell by 23 percent and the money supply shrank by 34 percent.
So van Buren wasn't re-elected, but that may simply be due to unfair perceptions and public sentiment in spite of the recovering economy, rather than any real problem with his policies. In any case it didn't turn out like the Great Depression, where the economic downturn dragged on for over a decade.
The free market improves the welfare of a few.
History says otherwise. The free market improves the welfare of the vast majority. A few benefit more than average, and a few benefit less, but the effect on the median is positive.
Without the market most of the people alive today would be dead of starvation, with the remaining few engaged in subsistence farming. The only people who aren't better off are those very few incapable of participating, in general due to a severe physical or mental handicap. Even those considered very poor are better off with the market than they would be without it.
Government ... can use fiscal policy to bypass central planners
Fiscal policy is central planning. It amounts to price controls on money, which have far-reaching effects throughout the economy.