Promotion/relegation doesn’t work like that. Indeed, the fact that the NFL is a group of franchises in a league that is unconnected to anything else is exactly why promotion/relegation won’t work in the US.
The only major professional sport in the US that would easily be set up for promotion/relegation is baseball. But to do it, we’d have to scrap the idea of farm clubs. I don’t see that happening.
Among the minor professional leagues, hockey and soccer could do it, though hockey farms and would have to give that up. But MLS can’t promote/relegate, given its single entity structure, and the NHL has the same sort of “franchise” concept going on that stands in the way of the NFL doing it.
Something similar to farm teams does exist in European soccer. In Spain, for example, the major clubs have B and sometimes C teams that compete at lower levels alongside regular clubs, and, if memory serves, are ineligible for promotion.
There are also some slightly shady relationships between major and minor clubs, wherein the big club often seems to sign players from the minnow. I don’t think this kind of arrangement is strictly allowed by the authorities.