E2E failing consistently

[aaemnnosttv]

Bug Description

All E2E jobs are failing consistently due to a few different stability regressions introduced recently.

Additional Context

See https://github.com/google/site-kit-wp/actions/workflows/e2e-tests.yml

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Acceptance criteria

• E2E workflows should be restored to a consistent passing state

Implementation Brief

• Merge Fix broken E2E tests #13122

Test Coverage

• No additions/subtractions

QA Brief

• No QA needed beyond review in CR

Changelog entry

• N/A

[tofumatt]

Moving directly to Approval as no QA is needed for this issue.