I've recently installed several nVidia drivers (from 5.32 to 45.23) on a VIA MVP3 board with DirectX 7.0a on Win98SE with a Geforce2 MX. All the versions I tried in that range worked without any malfunctions (other than some being slow). I don't think the GTS' compatibility would be any different, and your chipset is a newer generation than mine.
Have you tried getting other AGP cards to be detected correctly on this install, or even on previous installs with this board?
This sounds awfully much to me like a problem with the VIA AGP driver. It's reminiscent of problems I remember having on my MVP3 chipset when it's AGP isn't set up right.
Geforce (and maybe all nVidia) cards require the AGP support to be working correctly. Some other early AGP cards don't (3dfx, maybe others) because they don't utilize AGP features.
I don't know anything about that board, but maybe there are some BIOS settings (visible, or possibly not visible) that are causing AGP to misbehave.
If there's an option for the AGP speed, try turning it down. If the max is AGP 4X then set it to 2X, or even slower if you want to experiment. On VIA chipsets, the max supported AGP speed isn't necessary or useful, it's just there to break things.
If you're at a loss, then I'd try resetting the CMOS and then loading defaults. Occasionally the CMOS settings on a motherboard can get corrupted in a way that's not visible in the setup menu. There are hidden settings that the user never sees, so a CMOS reset can ensure those are not messed up. This is grasping at straws, but it's a possibility.
On my super-7 board, the BIOS settings weren't corrupt, but there was a BIOS setting that had to be set a certain way for AGP to work, even though it appeared to have nothing to do with AGP. I don't think your chipset is quite as braindead as my MVP3, but still, it reminds me of that experience.
I'm not sure how critical the order of operations is, but AGP needs to first be in a working configuration at the BIOS/hardware level, then the VIA AGP driver (included with the "4-in-1" drivers) needs to be successfully installed, and finally the nVidia driver after that.