Validate testing of headed vs headless browsers

In #135 (comment 1338494053) we noted that issues specific to desktop/headed mode were not picked up by CI tests.

In the job output of the "Headed *" jobs, we can see the version data says it's a headless browser:

$ curl -s http://localhost:9222/json/version
{
   "Browser": "HeadlessChrome/111.0.5563.146",
   "Protocol-Version": "1.3",
   "User-Agent": "Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) HeadlessChrome/111.0.5563.146 Safari/537.36",
   "V8-Version": "11.1.277.17",
   "WebKit-Version": "537.36 (@2b5ec366bb5b68dacca9719d5d83cb7e61fb6431)",
   "webSocketDebuggerUrl": "ws://localhost:9222/devtools/browser/2b67c447-77d5-4599-b003-e095736e29ca"
}

Would be good to validate whether we can connect to headed mode properly to test.

Bug from #135 (closed) should be reproducible for that purpose.