[ci] Centralize cache (#32672)

To avoid race conditions where multiple jobs try to write to the same
cache, we now centralize saving the cache and then reusing it in every
subsequent job.
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[//]: # (BEGIN SAPLING FOOTER)
Stack created with [Sapling](https://sapling-scm.com). Best reviewed
with [ReviewStack](https://reviewstack.dev/facebook/react/pull/32672).
* #32675
* #32674
* __->__ #32672

Author: poteto