You're spot on in pointing that out. There is one problem with that however.
The Federal Reserve system is directly supported by the economic output of the entire US and the taxing power of the US Government (it's further supported by the vast global demand for dollars and the dollar as the global reserve currency; that global reserve status can be further ensured by the US military). For example, if US budget deficits get too great, requiring support in the form of QE by the Fed, which then causes the dollar's value to drop, the US Congress can increase taxes to lower or eliminate the budget deficit. A healthy US economy is critical to supporting the Fed's abilities as it pertains to the USD. All of the Fed's actions and all of the things the Fed can buy, are directly linked to the US economy.
You can certainly print incredibly vast quantities of fiat - that won't help you at all if your economy is toast, if people don't fundamentally believe in what backs the currency, just ask Venezuela or Zimbabwe. When the Fed debases the dollar, what they're doing is taxing the wealth & economic output of the US economy, not solely manufacturing value out of nothing, they're spliting off value from an existing economy (and to a lesser extent global economic value from eg dollars & commodities held overseas).
Absolutely nothing like that exists for Bitcoin.