Yes, but it doesn't bolster your argument as much as you think. The state monad in Haskell offers a more principled (some say annoying) way of dealing with mutation.

IORef's if I recall correctly offer the type of mutation similar to what is colloquially thought of as mutation.

These add to your argument a bit, but they're usually seen more of a last resort

Sort of. My very opinionated point of view would be that Haskell demonstrates that the functional paradigm is the correct setting for the imperative style.