OMG. It IS the client. Nothing to do with OpenWRT. This issue happend so random, that my rule-out-strategy (by switching routers etc.) did not work out.
So after doing 15 min research with the right search terms I found out, that there indeed is a known problem with the e1000e kernel module, which is used for the Intel I219-LM.
So here are some resources:
Messages I saw:
e1000e 0000:00:1f.6: enp0s31f6: Cannot change link characteristics when SoL/IDER is active.
e1000e 0000:00:1f.6: enp0s31f6: (uninitialized): Reset blocked by ME
Lots of information about the problem:
The fix (at least for me):
rmmod e1000e && modprobe e1000e
reboot
The interesting part is, that after the reboot this issue is GONE. No need to re-apply any commands or the rmmod procedure. It's just gone.
Other attempts (did not work for me):
# slows down network speed
sudo ethtool -C enp0s31f6 rx-usecs 6000
# suggestion in the linked thread
echo on | sudo tee /sys/bus/pci/devices/0000\:00\:16.0/power/control
In my opinion something in the kernel / driver persists a setting / flag within the network cards internal storage that only gets reinitialized on complete power loss or reloading the kernel module.
The Fujitsu D3417 supports Intel AMT (Remote Management) that keeps the device "online", even if it is powered off. So I have a ping, can use meshcommander to connect to the device and power the system up remotely. Remote Desktop via VNC also works to navigate in the BIOS.
This AMT thing is tightly coupled to the Intel Management Engine and it seems that these two factors are interfering with the e1000e Kernel module (see the error message above). This is only a guess... but this bug / behaviour is very hard to catch...
There is also the fact that this never happend on debian 11 / Proxmox 7.3, so maybe the new kernel has some changes that made this bug happen again (regression).
However, I'll keep doing my research and as soon as I'm confident I found a solution, I'll post this here. Maybe I'll also file a bug in the debian / promox issue tracker.
Thank you all for your effort. I hope that anyone having this bug will find my explanation helpful and can workaround the problem.