When it comes to time travel stories (in any context, not just Trek), I've found it's best to simply enjoy them (if they're entertaining) and ignore all the implications. Otherwise, it just sucks the enjoyment out of the story.
It used to bother me (in my adolescent days) when time travel was used inconsistently within a particular "universe" (comics, movies, TV series, whatever) and when implications like some mentioned in this thread were ignored. Now, not so much. Probably because, over the years (adolescence was a long time ago for me), I've become accustomed to seeking immediate enjoyment from pop culture (any extra "thinking" it provokes is a bonus, not a requirement) and looking to other things for actual thought provoking discussion or contemplation. YMMV