Cons are that there is no incentive for some team owners to field competitive teams, especially when so much revenue is tied to league revenues or long term broadcast rights.

Pro is that the American sports with giant stadiums, conferences and division alignments and such would all make a relegation system hard to implement logistically.

Say the NFL were to relegate the Texans and Bears at the end of the season, there is no second tier to send them to / from which to promote new teams into NFL. Even in sports with minor leagues, those teams are owned by major league teams to develop talent.