Add e2e tests (likely via Playwright) #80
Description
What problem does this address?
The plugin currently includes PHPUnit tests, but let's consider adding in e2e / acceptance tests leveraging Playwright (as that's what WordPress core and Gutenberg leverage) to test the normal user experience within the plugin.
What is your proposed solution?
Leverage any existing utilities from WordPress core and/or Gutenberg to build out e2e tests (potentially added into the existing Test workflow and running in parallel to other tests there).
Also ensure we add a badge for the GitHub Action to the README.md file to quickly, clearly show that e2e tests are passing (or, sadly, failing).
