The only sport this could possibly work in America is baseball. I think it would be interesting. Even then you'd have to completely alter the relationships between the big league and the minor leagues and some more stadiums would need to be built when promotion did happen.
How could that work for baseball? There are already baseball teams in most of the markets able to support a team in a sport where you need to be able to draw tens of thousands of people 81 days a year (the home half of a 162 game season). Who would spend the vast sums of money building new stadiums in major markets would cost if you can't even guarantee that the team would be a major league club? That'd be crazy! And just covering other possible cities that don't have teams but could is something better suited for potential future MLB expansion, not relegation-system stuff (cities like Portland Oregon, Montreal, maybe something in Mexico someday, maybe a third team in New York, Las Vegas, etc.).
There are other reasons beyond money.
The logistics would be basically impossible, for one. England is the size of Alabama. Looking at the English football league set up, the top five are nationwide. Try having a fifth tier league in the US be nationwide - it's a practical nightmare. Where are you going to get the money to fly teams around? English teams can hop on a bus and be done. Heck, they can drive themselves.
You make a great point here about distances. All of the minor leagues in baseball, for instance, are regional, because there's only so far you can go on a bus, and private jets are out of reach for leagues that aren't rich, and minor leagues are not rich, and nor would be borderline clubs in a relegation system. It wouldn't work, for that reason as well as the stadium problem and maybe even above it.
So once you realise that it's practically impossible to have nationwide lower tier leagues, you start to hit the problem of exactly where the teams get relegated once they go down.
On top of this, the very nature of US sports (especially basketball and American football) make it incredibly hard to compete at a top level without the benefits that come with being at the top level. It's much easier (relatively speaking) for a lower tier soccer/football side compete against the top level - you can choose to stack your defence and counterattack, nullifying the skill/fitness advantage of your opponents. So when a lower tier team gets promoted to the top level, they're still able to compete to a certain degree. That's not going to happen in American sports.
Hah, as if those borderline relegation teams ever actually win championships...
Simply stating that money is the reason presents this idea that American sports owners are somehow more money driven than European football clubs, which doesn't really seem to be the case. Any league with private ownership is going to be money focused to a certain degree.
Yeah, there's certainly lots of big money in the European soccer system too, that's for sure.
I think the only shot would be college football between the big five conferences and the mid majors.
You actually sort of do have this in college sports, now that you mention it, with how schools switch to bigger conferences as their teams get more successful at one of the major college sports... it's not an automated relegation process, but the results are somewhat similar.
As others have said, money.
No, that's not the only reason.
Bad example. Manchester United (as much as I hate them) have virtually no risk of relegation. Sure, its hardly risk free, but they're as close to it as any team in a league with promotion/relegation. Obviously the Glazers are in it for money. And they've made a LOT of it from Manchester United. Would they make more of it if the league suddenly cut off promotion and relegation? Probably not. But that's not their problem.
No, they benefit from the relegation system because if you don't have relegation, then you get more pressure for competitive balance within the league. This has been a major focus in most American pro sports for decades now -- things like revenue sharing, salary caps, etc. as I said, that give all teams in each league a chance. Obviously Man U would never want something like that, they want to dominate their league most of the time and put a lot of their focus into Champions League stuff.
The best comparison of an American major league, in Europe, would be one continent-wide league of only a few teams from each country, as a set major league. That'd be a European counterpart to the American system -- the leagues within each country are too small in scale to compare.