Backport of run integration tests on push in main and release/* into release/1.15.x #21670
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Backport
This PR is manually generated from #21666.
The below text is copied from the body of the original PR.
Description
Our branch protection rules expect this...and we have no reason not to run these in release branches and on main.
Will manually backport to 1.15, 1.17, and 1.18.
Overview of commits