I disagree with the bolded part of your post.  Presidents can do a lot to bring down the cost of living without the Fed printing money.  Trump did many such things during his administration.

He increased energy production by opening up Federal land to drilling and fracking.  That lead to lower fuel cost for gas, heating oil, and natural gas, the cost of raw materials.  He got passed a tax cut that put more money into the consumer's pocket.  He brought back manufacturing to the US that increased the GDP and increased employment.  Billions in offshore monies were brought back into the country.