New Mac mini not recognizing generic Thunderbolt 3 to 2 display adapter (Apple 27" Thunderbolt Display)

Does either adapter work? Mine works fine however it is possible that the TB display all in one (octopus) cable is bad. In that case you could get a TB cable and connect to one of the TB ports on the display directly. I would only use an Apple official adapter as the system is very finicky on connection integrity.

This is a Thunderbolt Display – model A1407?

The Apple TB3-to-2 adapter will not present a plain DisplayPort signal on its TB2 side, so if you were plugging in a 27" LED Cinema Display – model A1316 – you would not get a signal.

Are you sure the black eBay adapter connectors are Thunderbolt (Lightning bolt icon on the plug)? There are adapters with similar looking connectors that won't with Thunderbolt displays. Their plugs may be USB-C but not Thunderbolt (no Lightning bolt icon), and the sockects may be Mini DisplayPort. The adapter's USB-C plug will fit a computer's Thunderbolt 3 port, and a Thunderbolt display's Thunderbolt 2 plug will fit the adapter's Mini DisplayPort, but it won't work.

I don't see any Lightning bolt icons on the plugs in the eBay photos of the adapters.

Here is a black Thunderbolt 3 to Thunderbolt 2 Adapter that does have a Lightning bolt icon on its plug.

https://www.startech.com//usb-hubs/tbt3tbtadap

It specifically says it works with the Apple Thunderbolt display.

Apple only makes the one model. I have found that using non Apple TB adapters is always risky.