Did you get your money back? Clearly they didn’t have a clue!
As the MagSafe does have power (the MagSafe cord is either Amber or Green) ideally you should monitor CoconutBattery to see what’s happening.
As the system doesn’t see the display could be a ThunderBolt cable issue or a flakey logic board connector. As you didn’t tell us what your system is I can’t dive any deeper in this direction.
How is the cable? Any noticeable damage or the cable is a bit sloppy within the connection? If you have a spare TB cable you can also use one of the displays ports to feed the video from your system (by passing the attached cable) does that work?
I would try a second system if you can or if your system has a second port try that as well.
If we can remove the cable and the system then this gets into the logic board within the TB display. It could be a cold solder joint or failing component.
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