I "fixed" it by finally managing to install the r8168 driver. But after a reboot, networking was totally dead. I haven't had time to do any further debugging.
Discussion
Q1: Have you installed "pve-headers" before the "r8169-dkms" package?
Q2: In the folder "/etc/modprobe.d" is a "r8168-dkms.conf"
Uncomment the last line and reboot this fix worked for me. but network is still unstable, so I will try the 5.15 kernel.