Playwright coverage for the public /status page

## Task

`public/status/` polls `/api/v1/health` and `/api/v1/ready`. Browser suite never opens `/status`.

## Files

- `public/status/index.html`, `public/status/app.js`
- `tests/browser/test_critical_paths.py`

## Acceptance criteria

- [ ] Test visits `/status`
- [ ] Asserts overall status plus health/ready payloads or check rows (database, redis)
- [ ] Does not flake on the 30s refresh (disable/stub interval if needed)
- [ ] `make test-browser` green

Start with public/status/index.html and public/status/app.js to understand the page's status and refresh behavior, then read tests/browser/test_critical_paths.py. Add a browser test that visits /status and verifies the overall status plus health/ready results or database and redis rows. Run make test-browser and ensure the test remains stable despite the 30-second refresh.

Written by the indexing model from the issue text.