First, US sports didn’t grow quite as organically as the European leagues did, where each city/neighborhood had a team, and they sort of naturally sifted themselves into a hierarchy of good to bad independent teams. This naturally lends itself to promotion and relegation, as each team’s independent of the others, and can move up or down in the hierarchy without regard to the others.
US minor league systems tend to either be centered around college teams (football/basketball), or are somehow tied to the top level teams. This is either by the entire team (minor league baseball and hockey) or individual players (NBA D-league). There are independent teams in minor league hockey, but they’re relatively rare.
What this means is that the whole promotion/relegation scheme wouldn’t work, as most minor league teams are arranged in a vertical player development chain upward to the major league team. The teams’ purposes are player development for the next step up, not necessarily to win in their own leagues.
For example, the Houston Astros (major league team) are affiliated with the following teams:
AAA - Fresno Grizzlies
AA - Corpus Christi Hooks
Advanced A - Lancaster (CA) Jethawks
A - Quad Cities River Bandits
Short Season A - Tri-City ValleyCats
Rookie: Greenville Astros, GCL Astros, DSL Astros
At each level, the major league team has personnel control, and can move a player from say… one of their rookie league teams to their AAA affiliate at their whim. Or for that matter, send a player from the major league team to AA or AAA ball to say… rehabilitate from injury without taking up a roster slot on the major league team.
Contrast this with say… Venezia in the Italian Lega Pro- they aren’t affiliated with any of the Serie A teams, and as such, could do well and get promoted to Serie B and then Serie A without affecting any kind of team affiliations.
It’s really the farm team systems that would likely be the biggest impediment to a promotion/relegation type system in baseball; since it’s all centered around the major league teams, it doesn’t really help anyone.