The spec might not even contain the level of detail necessary for that to be possible. That possibility is what makes modeling easier than implementing the spec.
Through a process of manual refinement, you can derive an implementation from the spec and check every step with the checker, but that's is more work than implementing the formal spec manually and is an most likely an overkill in practice.
People that can write proofs in Coq and Isabelle might prefer that over the TLC approach of exhaustively checking all the possible states allowed by a TLA+ spec. But writing proofs in Coq/Isabelle/Lean/HOL is a more sophisticated skill and requires even more training on the multiple theories available in these provers. The brute force approach of TLA+/TLC is more dependable as part of an engineering process. Some specs can be really hard to prove, but easy to exhaustively check.
Start looking for small opportunities to do something different and bold. Do that before you're asked to do it. Sooner or later you will get "hired" for the kind of job where you get to do the things you described here. That's how everyone else did it.
Through a process of manual refinement, you can derive an implementation from the spec and check every step with the checker, but that's is more work than implementing the formal spec manually and is an most likely an overkill in practice.