Majority rule rejected by a majority

When discussing voting method reform, we often talk about concepts like the majority criterion, which requires that “If one candidate is ranked first by a majority (more than 50%) of voters, then that candidate must win”.

This seems like an obviously desirable quality, if you grew up with “democracy” being treated as a synonym for “majority rule”, but some people (like me) believe that it’s possible to do better than majority rule, which means that “failing” the majority criterion can actually be a good thing.