@piotr86pl
WoL works reliably here with r8169. To enable WoL on boot basically every network manager (systemd-integrated, netctl, etc.) offers a config option.
For systemd see here
The mentioned 13GB/min would be more than 200MB/s, exceeding the rate of a 1Gbps link. So this number may refer to a compressed image. More relevant would be iperf results for assessing network performance. r8168 and r8169 both can go at full link speed.
To check for issues specific to the Realtek network chip version on your system (r8169 supports ~ 50 chip versions) a full dmesg log would be helpful.