Add E2E and acceptance tests for settings

[justlevine]

What problem does this address?

Gutenberg likes to break things, but we need to make sure updates don't break any functionality or extensibility.

What is your proposed solution?

We should implement codecept acceptance tests and playright e2e tests to ensure a stable backend.

What alternatives have you considered?

No response

Additional Context

Inspiration: WordPress/gutenberg#38851

[dpandey62]

Hi Maintainers! 👋

I’m a first-time contributor and would love to work on this issue.
Could you please assign it to me if it’s still available?

Thanks! 🙌

[justlevine]

Hey @dpandey62 thanks so much for your offer to contribute 🙌
No need to get "assigned" to something in this repo to pick it. Especially something like backfilling Playwright tests, which will probably be done best as several PRs.

Looking forward to reviewing your contributions