Build/Test Tools: Switch from Puppeteer to Playwright for the QUnit test runner.

This brings the QUnit tests inline with the e2e and performance tests that switched to Playwright in #59517.

This removes the `puppeteer` dependency, but `puppeteer-core` remains a transitive dependency due to `wp-scripts` and `lighthouse`.

See #64894


git-svn-id: https://develop.svn.wordpress.org/trunk@62411 602fd350-edb4-49c9-b593-d223f7449a82