While epsilon transitions often, but not always, cause this issue of not being able to complement the language by just flipping final/non-final states, it isn't always the case. For example, you can make an ambiguous NFA with no epsilon transitions by just doing epsilon removal, and then you still have the same problem but no non-trivial epsilon-reachable states!

On the other hand, you can have an unambiguous [1] NFA (for example, a DFA, but you insert some dummy epsilon transitions between a split up state) where you can just flip the final/non-final states and complement the language.

So, in the end, complementing languages described by NFAs without determinizing them first is a bit of a tricky problem.

[1]: https://en.wikipedia.org/wiki/Unambiguous_finite_automaton --- a superset of DFAs, but they can have epsilon transitions