refactor: use healthcheck path during test ping

What does this MR do?

We always send a ping request when setting up Pages during acceptance test to wait for the server to start.

This MR refactors the method to use the status path, avoiding side effects (ratelimit count, domain resolution, domain caching, etc.)

Blocked by fix: serve 404 if ctx is canceled while resolvi... (!796 - merged)

TODO

• I added the Changelog trailer (e.g. Changelog: added) to the commits that need to be included in the changelog
• I added unit tests or they are not required
• I added acceptance tests or they are not required
• I added documentation (or it's not required)
• I followed code review guidelines
• I followed Go Style guidelines