So I plugged in the Mac mini set it up with monitor over HDMI and then tried to connect the new adaptor and cable to one of the two Thunderbolt 4 ports (USB-C type ports on the back of the Mac mini) nothing happened. After watching several videos and consulting with them at the apple store I brought home a Thunderbolt 3 to thunderbolt 2 adapter and a. Well Interesting enough, I have a late 2013 iMac, and I got a Mac mini today. I think it is due to the screen recording being at 25fps and doesn't show it as smooth as it really is and hence my estimate of 30+. I did make a video but it looks more choppy than it really is. You do get the added bonus of having the Mac Mini in a window or full screen, very fast file sharing, shared keyboard, mouse and track pad and this should work with any mac not just those that used to have TDM Lower the display resolutions for higher frame rate.
Whilst not having the same screen refresh rate, it really is not too far behind at 30-50 if using adaptive 30+ for full quality rather than 60 (its not perfect). Both have good internet connection whilst having a good frame rate on the screen sharing. Seems to remember it after the first time too. Then when the connection and screens are shared turn on wifi and it connects to the internet via wifi and thunderbolt for the screen sharing. Just so that it picks up the thunderbolt connection first, I turned off wifi on both before trying to connect.