I have used r8168 NICs on Asus motherboards for many years and they usually worked no problem with either driver. But even Asus boards do not work well with incorrect driver with recent kernels
However, Gigabyte r8168 NIC motherboards don’t usually work well, if at all, with r8169 drivers
Internet is 3 times slower on Linux than windows in the same machine
IMO for this situation it would be better to try the quick and simple thing first before changing drivers and removing additional cards