[ci] Don't use third party action to push commits (#32648)

In light of recent third party actions being compromised, let's just
push the commit ourselves rather than use a third party action. We
already detect if changes are needed, so the step will only run if so.

I also added a `dry_run` option to the manual runs of this workflow for
testing.
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[//]: # (BEGIN SAPLING FOOTER)
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with [ReviewStack](https://reviewstack.dev/facebook/react/pull/32648).
* #32650
* #32649
* __->__ #32648