Can we try to extend from the bridge to a website and learn if the website is reachable?
Ticket #1851 (moved) suggests reprogramming bridges to connect to baidu periodically, and then report the reachability results in their extrainfo descriptors. We'd have to specify how that reporting works, how often the tests happen, whether we should generalize to write "baidu.com" in the consensus, etc.
A much simpler approach would be to build a circuit to the bridge, extend that circuit to baidu.com:80, and then look at the failure error code and discern whether it was reachable and just didn't TLS right or whether it was unreachable.
Can we manipulate Tor's protocol to learn this? Or is the error code too opaque? If it works, we should whip up a Torflow like thing that checks reachability of baidu from every bridge.