test(docs-infra): avoid running expensive operations when focusing/skipping tests (#43963)

Previously, the pre-computation of branch version info for the
`deploy-to-firebase` tests would always take place even when the tests
that used that info were skipped. This pre-computation involves network
access and may add a noticeable delay.

This commit refactors the tests to not do the pre-computation if the
corresponding tests are skipped. This allows for a faster iteration
when focusing on a different set of tests.

PR Close #43963