https://aur.archlinux.org/cgit/aur.git/tree/PKGBUILD?h=r8168-dkms#n43
the build/package do blacklist r8 169 already.
But with the removal of the r8168 package for the default arch kernel, from the repository there is no working r8168 package as far as i know… only the one for LTS kernel is left at the repo.
And the r8168 modules are supporting the same cards as the newer r8169 shipped with the kernel.
So first make sure your device is really not working with r8169 module.
try:
sudo rmmod r8168
sudo modprobe r8169
sudo systemctl restart NetworkManager
And show yay -Qs r8168