How can I use both Wi-Fi and Ethernet on my Mac mini?

I have the same setup and having both on with ethernet as the top priority and have no issue. Mine do not cancel each other out. Not sure why you have that issue. You can also try to use a WIFI extender which are inexpensive. I have one in my basement and it helps the WIFI range. then perhaps you can bypass the ethernet connection.

Not sure either why you have this issue.

By any chance, do you have a dual band router (with 2 wifi settings?)

I used to live in a cottage with the mini in my studio in the basement, and had to ensure it was on the proper band (haf dual band in the house) when i needed to use airdrop (i too was on ethernet as priority).

It works for me. Do you have a similar setup:

All Macs have Ethernet set as 1st and Wi-Fi as 2nd in Network Settings Service order, with Wi-Fi set ON (location services, AirDrop need it to work). Desktop Macs connect via Ethernet and laptops might use either if the user bothers to connect the cable -- Wi-Fi speed is OK around the house and about 50 meters outside.

500 Mbps service and Ethernet Cat6 cabling in the house. One master Wi-Fi router + a small Ethernet switch and two otherwise identical remote slave Wi-Fi routers connected to the main Wi-Fi router via Ethernet.

All Wi-Fi routers have the same SSID name in their 2.4, 5 and 6 Ghz bands.

macOS Firewall disabled (I used to have it ON but I have been educated that it servers no real purpose).

If a similar setup does not work for you, then maybe you might try to delete and add back and re-configure Ethernet and Wi-Fi services. And maybe try Google's DNS servers.

if the wifi router is set to bridge 2.4Ghz wifi 5Ghz wifi and wired ethernet devices connected to each time will communicate without any issues