Consider automatic CI check for breaking API changes

[alamb]

Is your feature request related to a problem or challenge? Please describe what you are trying to do.
As part of the discussion about release cadence in #5368, what continually comes up is "when do we introduce breaking API changes" and "what effect does maintaining the check / discipline of avoiding the breakage have on maintainer bandwidth"

Specifically, carefully evaluating all PRs to see if they have breaking API changes is non trivial and error prone, and thus increases maintenance cost.

Describe the solution you'd like
I would like to implement some sort of automated CI check that verifies semver compatibility and flags the PR appropriately if there is an API change

Describe alternatives you've considered
Perhaps we can try using one of the following crates

• https://github.com/obi1kenobi/cargo-semver-checks
• https://github.com/Enselic/cargo-public-api

Additional context