Migrate E2E tests to Playwright

[acicovic]

Is your feature request related to a problem?

WP/Gutenberg are migrating towards Playwright for E2E tests.

Describe the solution you'd like

Migrate/convert all our existing E2E tests to Playwright.

Describe alternatives you've considered

Do nothing, but this doesn't seem like a viable approach. In the wordpress/e2e-test-utils package page, it is stated:

Note that there's currently an ongoing project to migrate E2E tests to Playwright instead. This package is deprecated and will only accept bug fixes until fully migrated.