With Nyx¹ (formerly tor-arm) you can see the IP, fingerprint and nickname of the Tor relays that form a cirquit. Middle node can see that a bridge is used, but not whether and which pluggable transport is used from the client (Tor-Browser) to the bridge.
¹Or other tools that use Stem interpreter
I don't want to show a productive relay. Hence: Tor Nyx on a ...
Would the next step be to request private bridges by e-mail?
Well you have two options to get bridges:
Email firstname.lastname@example.org through Gmail or Riseup.
Use MOAT to fetch bridges from the Tor browser. If you want to know how to use MOAT, check out: https://tb-manual.torproject.org/bridges/.
Also why does the tool only accept e-mail requests from ...
If any country censoring Tor, such as Iran or China, decided to host bridge relays, won't they be able to figure out where Tor users are and then could take action?
They can learn the IP addresses of Tor users that connect to their bridge relays. How easily they can identify a single person, find their location, and take disciplinary action probably depends ...