What is on that Thunderbolt 4 cable in the M1 Mac mini's other Thunderbolt/USB-C port..?
If it is another Display? Then your best bet, is to replace the old Mini DisplayPort Cinema Display with a 4K HDMI display.
In the mean time, if you decide to test the HDMI to Mini DisplayPort cable adapter that you linked above.
https://www.amazon.com/Elecbee-DisplayPort-Converter-Adapter-Female/dp/B0B185BTHC/ref=sr_1_5?
Then make sure that you can return it, if it does not work.
For reference, the Mi Mac mini only supports one on the Thunderbolt/USB-C port and one on the HDMI port.
go to > Connect a display to Mac mini - Apple Support select the 2020 and 2021 versions.
- For Mac mini with M1 chip, you can connect one external display up to 6K using a Thunderbolt 3 (USB-C) port, and one external display up to 4K using the HDMI 2.0 port.