Small update for those who’d be interested: in the ned I upgraded to another system, but with the same situation: the system provided the generic r8169 driver whereas the r8168 was more suitable.
In the end I got the latest official r8168 driver (which includes a very easy to use bash script to install) and had to disable safe boot mode and all was well. (I also tried the dkms which didn’t work out, but I seem to remember I might not have figured out that I needed to disable safe boot yet, so dkms might also work…)
Thanks for the good input!