They are controversial because supporting them raises matching time to O(2^n) in the worst case - which is dangerous - while not supporting them enables matching time to be O(n) in the worst case. And a user might not be aware of that danger.

Also, I think it's misleading to mention NFAs and DFAs alone, like there's no deeper reason. The deeper reason is that:

  Regexes (or NFAs, or DFAs) classify exactly the properties of strings which can be checked in constant space.

It turns out that any such string property can also be checked in O(n) time, and in one single left-to-right pass.

Briefly returning to backreferences, supporting them makes constant space unachievable. They also make linear time unachievable, since matching them becomes NP-COMPLETE.

Also, no regex implementation actually uses DFAs. NFAs yes, but DFAs no. The point of NFAs is that they're easy to execute in constant space and linear time, while also making concatenation and Kleene star easy to implement. It seems much harder to implement concatenation and Kleene star using most other constant-space automata - like DFAs - which is what actually forced people to invent NFAs.