From this, it's clear you've never read and understood Moggi's seminal paper. Monads are functors with some extra monoidal structure. The concept, and even Moggi's use of it in categorical semantics, has nothing to do with "worlds".
The important realization is that there are many more monads than just the one hardcoded into one's programming language of choice. State, error handling, parsing, reading from an environment, backtracking, nondeterminism, mutable state, logging, probability, continuations, async/await, I/O, ...these are all just specific instantiations of the general interface of monads. Recognizing that allows you to build abstractions that work for any monad, rather than re-discovering and re-implementing the same idea for each one separately. It's been a remarkably fruitful area of research.
You'll have to elucidate, I don't follow.
>uncomposable
Demonstrably false, see monad transformers.
>better left alone.
"Yeah, well, that's just, like, your opinion."
No, they’re not. That’s just an analogy that’s sometimes useful when describing them to imperative programmers to aid intuition.
I'm not backing down from the core claim that Moggi realised that CT monads are a nice formalization of side effects in pure functional programming, and that this caused a flurry of interest in "what else" of high impact/interest CT can explain of programming, and that there isn't much else.
No. The state and environment monads fall under that rubric, but many of the others are far removed from what you're describing. You can't implement backtracking in a purely functional way simply by passing around an extra parameter, because backtracking affects the control flow. Continuations are even stranger—they're not even algebraic!
> I'm not backing down from the core claim that Moggi realised that CT monads are a nice formalization of side effects in pure functional programming
No, Moggi's paper was about formalizing the semantics of impure programming languages. It wasn't until Wadler that we started using monads to model side effects in pure functional programming. Please read the actual research before citing it.