How to detect network connectivy on an interface?

If they are watching the screen why not go to System Preferences -> Network and watch the connected indicator. It will show connected when connected to a switch even before DHCP.

This is also where a cable tester comes in handy. Even the cheesiest of testers will work for this. Check these out:

https://www.homedepot.com/p/Klein-Tools-LAN-Explorer-Data-Cable-Tester-with-Remote-VDV526-100/309925931

Or a simple tone tester...

https://www.homedepot.com/p/Sperry-Lan-WireTracker-Tone-and-Probe-Wire-Tracer-ET64220/202520187

What if you try to share you Mac pros Ethernet connection (w/DHCP) to your WiFi card. Connect with your phone, it should connect and say no internet connection. Run off to the closet, plug something in then check your phone?

You could even set up ssh and log into your computer from your phone over WiFi and run ifconfig, ping, ect.