Immutability is an implementation detail with no inherent value. What we actually care about is referential transparency, encapsulation & tracking of effects, performance, abstraction, thread/type/memory safety, etc. The State monad in Haskell is implemented without mutation, but presents an API with mutation. The ST monad is implemented with mutation but presents a pure and referentially transparent external API.

It is fair to say that functional programming is primrarily in favor of immutable data structures because it is simpler to implement them efficiently in a setting where referential transparency is the default. Mutation (meaning referential intrasparency) is just more complicated, and we need heavier machinery to structure it nicely. We may use linear/substructural types, uniqueness types, regions, etc for this purpose, but here are a number of choices, and the picture is not as clear-cut as with plain pure lambda calculi. Just remember that mutation is not bad by itself, many functional programmers are often happy to speed up production code with mutation, it's just that it is usually more complicated to structure nicely.