You mean, like we elect congress, seems like a recipe for disaster.

Already, every American profits from whatever profits the FED. FED profits are paid to the treasury.

The Fed returned $76 billion to the Treasury last year, down from $109 billion in the prior year, according to the document, which updated figures first released in January.

https://www.bing.com/search?q=FED+profits+returned+to+treasury+last+year&form=ANNTH1&refig=8bfc1f3dbff941a1946e0843194722a9

You were not specific but maybe you meant that each person would get a personal check for their share of the FED profits. Sure, why not. Last year everyone could have a personal check for $200 ish from the FED instead of it going to the Treasury.