Could COVID vaccines facilitate the spread of mutations by biological evolutionary processes?

Darwin teaches us of the "survival of the fittest". COVID vaccines kill off standard COVID variants in a narrow spectrum, I believe, much narrower than would be the case for natural immunity. Thus, could COVID vaccines facilitate the spread of non-standard COVID mutations by killing off their competition, and, making these mutations the "fittest"? With all the standard forms of the virus eliminated, the mutations would have all biological resources -- that is, human victims -- at their exclusive disposal. Surely, this would make them more likely to survive, wouldn't it?