[dnm] [optimize] MRE::typ(): can we avoid recalculating types?

[mgree]

#30871 rebased onto #30881, to test a hypothesis of @ggevay.

Motivation

• This PR fixes a recognized bug.
  See [optimize] MRE::typ(): can we avoid recalculating types? #30871 and Sometimes we can give the type to MRE::take_safely #30881.

Checklist

• This PR has adequate test coverage / QA involvement has been duly considered. (trigger-ci for additional test/nightly runs)
• This PR has an associated up-to-date design doc, is a design doc (template), or is sufficiently small to not require a design.
• If this PR evolves an existing $T ⇔ Proto$T mapping (possibly in a backwards-incompatible way), then it is tagged with a T-proto label.
• If this PR will require changes to cloud orchestration or tests, there is a companion cloud PR to account for those changes that is tagged with the release-blocker label (example).
• If this PR includes major user-facing behavior changes, I have pinged the relevant PM to schedule a changelog post.