Even so, one might wonder why we don't try making systems that take different approaches. For example, after a traditional first pass of output, they could do sliding-window "optimizations" considering each token in the context of tokens both before and after, and possibly replace words or phrases in-place.

For example, I've noticed quite a few cases recently of LLMs outputting "but" where "and" would make more sense, or vice-versa. Surely that could be improved by such an approach?

People have and are trying things. Lots and lots of things. They just don't go around promoting failed ideas.

Look up diffusion models.