Closes https://github.com/microsoft/playwright/issues/32076.
This PR rewrites `watchMode.ts` to use `TestServer` under the hood. It's
essentially a complete rewrite, so don't pay too much attention on the
old implementation. Note that there's no changes to tests, so all
behaviour we have specced out there still works.
To make this work without a superfluous WebSocket connection, I had to
refactor `TestServerConnection` a little. Originally, I pulled this into
a [separate PR](https://github.com/microsoft/playwright/pull/32132), but
then realised how small the refactoring is. So it's in this PR now. Let
me know if you'd like to land it separately.
201bad75d3