I agree and would add that the UI is also a pile of mutable state. Even if you model the DOM using a pure function, there's still scroll position, selection, animation, history, and so on.
At the end of the day, users interact with state. We need languages and techniques that manage it well.
Yes. This is one of the core problems of FP style analysis of what's "wrong" with software development. Sometimes those people act like state is some sort of bad habit, like chewing tobacco. But the way they say to get rid of state really just hides it behind large state-management engines, like browsers and databases. It boils down to "state for thee but not for me". Well, great. But some of us have to deal with the fact that computers are often used to model the real world, which isn't a pure function.
At the end of the day, users interact with state. We need languages and techniques that manage it well.