Because of how other US sports are structured (Geographical Divisions, a draft, playoffs etc.) the UEFA structure of Promotion/Relegation would be a hard sell in the US (right now). Relegation for a team in any US league would be catastrophic from a financial standpoint. Look at baseball, as an example. Teams like the Astros, the Cubs, the Marlins, the Royals (although they're improving) would all be relegated, for arguments sake. Apart from the Cubs who have a historically loyal fan base despite not winning anything in over 100 years, it's not far-fetched to think the other clubs would collapse without enormous investment from an owner. People just don't go to watch a losing side in the US when they can still support them from home. Its a philosophical difference between the European supporter and the American supporter.
Now, obviously that's painting with a broad brush and not everyone feels that way. There are die-hard fans of every team out there, despite poor performance. But the American sports fan - and I'm willing to bet this applies to exactly zero people on this forum - is fickle (not necessarily with allegiance, but with support for a regime or player). We live in a "What have you done for me lately?" society of fans. If the answer to that question is "Well, you got relegated," especially in an arena as fragile as MLS & American soccer in general (and I say fragile in light of expansion bringing in new fans) it would be disastrous for the fan base and possibly the sport.
This is not to say that promotion/relegation can never work in MLS. At the moment, most MLS fans are familiar with the process from following various club leagues in other countries, which eases the burden for the current teams and in the current structure. However, in the state of MLS now - even if we adopted one of the other American leagues - the expansion coming over the next 10 years makes the idea precarious at best. No one expects an expansion team to win the cup in their first year (except of course for NYCFC who will obviously win ;-)) but to adopt a second league as the relegation league, then continue to expand the league to however many teams and introduce promotion/relegation during that process would be such a mess of teams moving all over the place in the tables that new fans would come in and say "Huh?" MLS would have to wait until they (and their prospective relegation league) had a stable base of teams, a relatively stable income across the board and a good academy structure to keep the lower tiers competitive on some level.
TL;DR: Promotion/Relegation isn't impossible in the future, but it will have to wait for MLS to be a way more stable league with a fixed amount of teams, financial stability and more widely (and more importantly, in my opinion) - soccer academies to take hold and grow in the US to make it possible for lower-tier teams to cultivate talent and remain competitive.
Now, obviously that's painting with a broad brush and not everyone feels that way. There are die-hard fans of every team out there, despite poor performance. But the American sports fan - and I'm willing to bet this applies to exactly zero people on this forum - is fickle (not necessarily with allegiance, but with support for a regime or player). We live in a "What have you done for me lately?" society of fans. If the answer to that question is "Well, you got relegated," especially in an arena as fragile as MLS & American soccer in general (and I say fragile in light of expansion bringing in new fans) it would be disastrous for the fan base and possibly the sport.
This is not to say that promotion/relegation can never work in MLS. At the moment, most MLS fans are familiar with the process from following various club leagues in other countries, which eases the burden for the current teams and in the current structure. However, in the state of MLS now - even if we adopted one of the other American leagues - the expansion coming over the next 10 years makes the idea precarious at best. No one expects an expansion team to win the cup in their first year (except of course for NYCFC who will obviously win ;-)) but to adopt a second league as the relegation league, then continue to expand the league to however many teams and introduce promotion/relegation during that process would be such a mess of teams moving all over the place in the tables that new fans would come in and say "Huh?" MLS would have to wait until they (and their prospective relegation league) had a stable base of teams, a relatively stable income across the board and a good academy structure to keep the lower tiers competitive on some level.
TL;DR: Promotion/Relegation isn't impossible in the future, but it will have to wait for MLS to be a way more stable league with a fixed amount of teams, financial stability and more widely (and more importantly, in my opinion) - soccer academies to take hold and grow in the US to make it possible for lower-tier teams to cultivate talent and remain competitive.