Thank you, Coius. I didn't mention it, but I have those drivers, and those are the ones that I tried. And the machine has 10.2.8 installed. That's what I've been running.
It probably wouldn't hurt for me to download them again...
When I install them again, or install the later version, will the installer just write over the older version?
The other night when I was trying this, I tried reinstalling probably about a dozen times, but even though the installer never hiccoughed, the question in my mind was whether it was actually overwriting the previous installation attempt. I tried the 10.2, 10.3 and Tiger drivers, but they all had the same effect, which is to say, nothing. There is a rtl8169.kext in the extensions folder, but other than that, no apparent effect.
I see that you got the 8169 to work in OSX. Have you ever gotten it to work with 10.2.8? Maybe their 10.2 driver worked with earlier versions of 10.2, but not with .8 and then when they wrote their 10.3 version, maybe that didn't work with 10.2.n. It seems unlikely, but it would explain why I can't get this stinking thing to work.
My main goal is to get a bunch of files off of the Beige G3, so since the OS9 installation is working, I think I'll just use that. There is a part of me that wants to argue with the machine until the 8169 works under OSX, but for my immediate needs, it just isn't necessary.