For this I turn to an old quote from college football, first used by the coach of Navy in the 50s, but more recently by Nebraska's Tom Osbome after he lost the 1984 Orange Bowl in the final minute on a decision to go for the win rather than the tie (when college games were able to tie, that rule changed in '96):
"A tie is like kissing your sister."
I'd love to see relegation, but I don't think the pro programs would do it as the system is very different creating a number of issues:
Legally speaking, the leagues are organized around the owners --they were created by the owners with the President of the leagues serving to the pleasure of the owners --only in rare cases, when the owners are against a single owner, would you find the league taking any negative action against a team. Any rule changes even vaguely like relegation would need to be voted on by team owners who would never put team profits on the line like that. While there are teams that generally do worse than others, even the worst team will sometimes have a surprise turn around. The incentive to make a team better is getting people to go to games and the generation of profits, otherwise a team will be sold or move.
The minor league (lower division) teams in something like baseball are farm teams that are often partially owned by the major league teams in their own "farm system". Also there is a massive difference in facilities between even the best minor league baseball teams and the pro teams.
American football wouldn't work because there just isn't a minor league --there have been attempts, but the closest comparison is College Football, which is university level.
About the only one that could theoretically pull it off would be basketball, but again --there's just too much of a financial gap and vested interest by the pro clubs.
The money difference between American pro sports and even the top of non-American sports, English Premiere League, is stunning. The last I checked, American College football has a bigger financial footprint than all other international leagues other then Premiere League.