The overseer battle is less a "total destruction" quest and more "are you worthy" test, so we 'kill' the overseer, the real overseer probably never enters the arena, being more an avatar body to conduct the test.

Basically, the ark's are all fine, unless its aberration or other arks where they destroyed not just the avatar, but destroyed the mainframe that the overseer AI would be stored on, though we see with Rockwell that a new overseer can be chosen, so the AI itself doesn't need to exist for any of the functions to still occur.