I have the impression that Blackmagic is deliberately making life more difficult with the Mini.
Yes, Ethernet is possible, directly to the PC or via a switch.
The difference: you have to configure the IP address(es) manually, whereas Pro/Extreme obtain the IP address via DHCP.
This means that you first have to check what IP the PC has and then configure the ATEM mini accordingly.
Page 103 in the manual:
https://documents.blackmagicdesign.com/UserManual...

If you only want to control the ATEM Mini via the ATEM software, a USB cable (which is connected anyway if you use it as a "webcam") is sufficient. I can only recommend the software, it really brings a lot of additional benefits!