All problems solved, so for completeness (in case anyone else has a similar problem):
Turns out, the MAC address was messed up ("FF FF FF FF FF FF") when the BIOS flash went wrong or during the recovery process, causing the LAN adapter to not work until it gets assigned a correct MAC address.
Interestingly Windows automatically set the correct MAC address when I removed the adapter via device manager and installed it again. This was actually a bit confusing while trying to pinpoint the problem, but I guess it must have something to do with the fact that Windows already knew the correct value before the BIOS flash went wrong, therefor being able to asign the correct MAC address again.
To fix the MAC address permanently, I used the EEPROM/EFUSE Programming Utility for Realtek RTL8168 Family Ethernet Controller (PG8168.EXE).
PG8168.exe /efuse /barmac
After that I simply entered the desired MAC address and it was flashed successfully.
Now the MAC address is correct and also survives a power loss.
Note: The mentioned tool (PG8168.EXE) works for the Realtek 8168 family only, so if you have another LAN adapter, you need to get the appropriate programming software for this specific model.