They're "capitalists" as long as they have a thumb on the scale, but they're not really capitalists who would cheer a true free market in currency.
So it's largely the market that drives short and medium term inflation through borrowing. The Fed uses treasury bonds to balance the money supply.
It's not a perfect system, but it's the best system we know of.
Nothing about the USD is free market.
The USD is shored up by the petro dollar standard and its status as a reserve currency esp. in the commodities market and its supply and management is governed by a quasi-private institution that is in bed with Wall St.