Add to the pull request checks the step that failed during the last release

During today's Python SIG meeting we discussed the most recent release, which was successful in the otel-python repo but not in contrib. As a result, some new installations were failing.

To reduce the chances of a surprise failure while releasing, this issue aims to track the effort to ensure that we are exercising the same checks for PRs that we effectively have for releases -- we may need to add one corresponding to the failure during the last release.

Is your feature request related to a problem?
After the last release, some new installations were not successful.

Describe the solution you'd like
Sync the explicit checks that run when PR is submitted/updated with the effective checks that happen during a release.

Describe alternatives you've considered
It's possible that the failure was spurious and that the check is already happening in which case, there may be nothing to do.

Additional context
cc @lzchen as he may have additional details

Compare the explicit checks run when pull requests are submitted or updated with the effective checks performed during releases in this repository. Investigate whether the release failure is already covered or identify the missing check; done means PR checks exercise the relevant release checks and reduce the chance of a repeat installation failure.