Yes, conferences have this choice (or at least used to, since as someone pointed out, the Ivy League used to use regular season record).
I didn't mean to imply that conferences are forced to have a tournament, just commenting that the randomness of auto-bids for conference tournament winners (and the increase in number of conferences) leads to a weaker pool. I acknowledge that conference tournaments are exciting and bring revenue, but there are other ways to handle this, like giving autobids to both the regular season winner and tournament champion and increasing the number of play-in games to accommodate the worst ones without taking the place of at-large selections.
Edit: I don't like play-in games for 10 seeds. Make all the really bad teams play each other for a real spot