Hello!
Eddie should allow the execution of binaries when they are not owned by root, only if they pass the checksum verification. If they don't, only files owned by root will be run with root privileges. The files packaged with Eddie should pass the checksum verification. Why it does not happen is a matter of an ongoing investigation by the developers.
With that said, the system refusal to change ownership even to the superuser is probably caused by a terminal that's not allowed to have full disk access, according to System Integrity Protection. You should grant Terminal program full disk access in the following way:
Make sure that Terminal is not running.
Open System Preferences > Security & Privacy from the Apple menu..
Click the padlock icon at one corner of the window and enter your administrator password to make changes.
Go to the Privacy tab.
Click the [+] button at the bottom of the list on the right.
Navigate to Terminal program location in the Applications folder ([...]/Applications/Utilities/Terminal).
Add Terminal to the list and make sure that its checkbox is ticked.
Then, open the terminal and try again to change ownership of the relevant files
sudo chown root <full path and name of the file>
You will need to change ownership of both wg and wireguard-go files.
Kind regards