Add publish-check-compile workflow (#6556)

Add publish-check-compile workflow

This Applies staged prdocs then configures crate deps to pull from
crates.io for our already published crates and local paths for
things to be published. Then runs cargo check on the result.

This results in a build state consitent with that of publish time and
should catch compile errors that we would of otherwise ran into mid
pubish.

This acts as a supplement to the check-semver job. check-semver works on
a high level and judges what changes are incorrect and why. This job
just runs the change, sees if it compiles, and if not spits out
a compile error.