> So first let it translate natural language to a formal language, from there allow it to use a logic engine to make verifiable transformations (correctness-preserving), and finally translate back to natural language.

Linguists in the Richard Montague tradition have indeed attempted to use tools like formal logic, lambda calculus, continuations, monads, modalities etc. to try and understand the semantics of natural language in a way that's both logical/formal and compositional - i.e. accounting at least partially for the "deep" syntax of natural language itself, such that a fragment can be said to have a semantics of its own and the global semantics of a broader construction arises from "composing" these narrower semantics in a reasonably straightforward way.

This is pretty much the same as trying to take the "let's translate natural language into formal logic" proof-of-concept exercises from a text like OP (or from your average logic textbook) seriously and extending them to natural language as a whole. It turns out that this is really, really hard, because natural language mixes multiple "modalities" together in what looks like a very ad-hoc way. We only barely have the tools in formal logic to try and replicate this, such as continuations, modalities and monads. (Linguists actually talk about many phenomena of this kind, talking about "modalities" is just one example that's both general enough to give a broad idea and happens to be straightforward enough on the logical side. You have quantification, intensionality, anaphora, scope, presupposition, modality proper, discourse-level inference, pragmatics, ellipsis, indexicals, speech acts, etc. etc. etc.)

And because the semantics of natural language is both so general and so hard to pin down, it doesn't seem useful to "reason" about the logical semantics of natural languages so directly. You can of course use logical/mathematical modeling to address all sorts of problems, but this doesn't occur via a verbatim "translation" from some specific language utterance.