I have 3 ASUS P8Z68 boards and a P8Z77 board all using the R8169 driver for the motherboard Ethernet port. All have additional network cards. The ones with additional network cards using Realtek components also use the R8169 driver. They are running Slack64 13.37 and Slack64 14.0 with the generic kernel.
I have a non-simple network setup with multiple LANS, VMs, special firewall rules, and such running in production for the last year. So I can't say with 100% certainty that the motherboard Ethernet port using the R8169 driver that comes with Slackware can access the Internet through an external router. Too many other components are involved in my setup.
If you're using an external router with a simple out-of-the box Linux setup I can't see why the driver would impact the router's ability to route packets.
I did have a great deal of difficulty getting the P8Z68 boards to "cooperate" regarding video and other basics (such as booting
). It took me several weeks to get things working. I recall obtaining a driver from Realtek during this process but I don't remember if it was needed in the final setup. My configuration notes are not available to me right now.
What finally solved the issues for me was a BIOS update that ASUS made available during this time period. You may want to make sure you're running a current BIOS on the ASUS board.