Ethernet stops working every 1-2 days

Self-assigned IP addresses means the DHCP server (in the router) is not reachable, or is not working.

Remove the add-on first-few-hops VPN client entirely, ans any add-on cleaners or add-on security apps, and restart.

Replace the Ethernet cable. (Cheap test, and possibly cheap fix. Bad cables can cause problems.)

Check for firmware updates for the router, and apply the newest.

Quite possibly where this is all headed: acquire a replacement Wi-Fi 5 or Wi-Fi 6 router. They do fail.

You said

The ethernet will go into the "self-assigned ip" state and will not work again until I power cycle the router.

self assigned ip address is telling you the router is not responding back with an ip address. Could be your router has limit set for number of con-current connections.

Or it could be your dhcp time limit is set for 48hrs and it has expired and failed to get a new address from router due to a bug.

you also said

It doesn't help that my router has some sort of bug right now (Netgear Nighthawk) that will not allow me to assign a fixed (local) ip address for my Mac Studio.

Check to make sure the router Sw is up to date. The fact that it won’t let you assign ip address, Manually is a problem to. If it’s updated time to call Netgear for support or purchase another vendors router.

oh on the VPN, get rid of it for testing reasons

VPN is very important tool when your Institution (employer or educational institution) chooses to require it. In that case, they also offer some modest support to help you 'get with the program'.

The general advice is that it is a serious mistake to use a VPN that you added yourself. Such a VPN does NOT improve your security. You are already encrypting your over-the-air connection to your Router. Your Router uses Network Address Translation and a built-in state-wise firewall to hide your local IP address from exposure on the greater internet. And every important web site you visit uses httpS: encryption.

Use of a VPN reduces your overall internet performance, and many of these packages choose to branch out and do more invasive things like read ALL your files, non-stop, looking for antique Windows virus patterns in your graphics files. Its complexity is daunting, and it causes seemingly unrelated problems.

Step one to get better stability is to remove that VPN software, completely, at least while you continue to debug.