Why? If you can just write whatever you need in the language purely itself, why would you want to do that. Please be detailed and try to keep your dictionary simple. Using words like "reified" doesn't really help anyone to understand what you're trying to say, when it's known and accepted that the topic is very complicated.
All of these things - Functors, Applicative, Monads, Lenses - are all design patterns in functional programming. In the same way you can use an Observer pattern in OOP, you can use a Monad pattern in functional programming. Neither adds neither computational nor expressive power to the language, but it does avoid code repetition.
(2) Monads cause you to tag your types with effects that they are using. This means that your code is self-documenting about the side-effects it performs. In non-pure languages you still have to trust that the code isn't breaking convention, but it's a very powerful documentation methodology.
(3) Creation of arbitrarily complex effects via stacking. While your language might have state and even non-determinism, it probably doesn't have their combination built-in. Their combination forms a very nice backtracking, non-deterministic parser almost automatically (see a loonnngg comment of mine elsewhere in this thread).
(4) Optimization and compilation. While not directly possible with vanilla State, once you begin to use monads to represent effects it's easy to go back through and optimize your entire "language" by optimizing your monad implementations. You can even optimize them right into other targets like the Accelerate library which cross-compiles certain fragments of your Haskell—which look like mostly garden-variety normal Haskell—into GPU code.
Even if you have mutable state you might want to use the State monad to implement it. Even if you have continuations you might want to use the Cont monad to implement it. No language has Parser built into the language itself—it's really just an invention available when you start to reify contexts as datatypes.
Monad is about abstracting over a common pattern of sequencing side effects. These can be available in the language or not. Once you reflect that sequencing into a monad then it often becomes reified in such a way that makes side effects easier to track and understand.