Both are from the same period as the motherboard, Hell my SE440BX-3 (Gateway OEM) shipped with a 16MB TNT.
I'd put it down to some hardware combinations just don't play nice

I have a similar issue with my GF4 MX440 when it is inserted on a Slot 1 AGP occasionally. Try twiddling the cards when the computer turned off a bit to see if the POST screen appears.

I think my SE440BX-2 may be just a little picky about its AGP cards, which is weird given its reputation for stability. It's possible the caps are failing, I guess. I pulled out my Socket A box, which has a real reputation for flakiness because of the VIA Apollo KT133 chipset, and it handled ALL of the AGP cards just fine. It may become my new test bench, oddly.

You might have dirty contacts in the board, making it flaky or picky about which cards it makes good contact with.
I've used Deoxit to clean a couple Slot-1s before and was convinced it made them more reliable. Unfortunately that stuff tends to be pretty ridiculously priced.
But if you've reseated those cards over and over and it always behaves the same, then it starts to seem like something else could be going on.

It's hard to think of what would make a Riva128 or TNT less compatible than a Geforce2 MX with a 440BX board. I doubt there's much difference in power draw.

shamino wrote on 2021-08-27, 01:17:

You might have dirty contacts in the board, making it flaky or picky about which cards it makes good contact with. I've used Deo […]
Show full quote

You might have dirty contacts in the board, making it flaky or picky about which cards it makes good contact with.
I've used Deoxit to clean a couple Slot-1s before and was convinced it made them more reliable. Unfortunately that stuff tends to be pretty ridiculously priced.
But if you've reseated those cards over and over and it always behaves the same, then it starts to seem like something else could be going on.

It's hard to think of what would make a Riva128 or TNT less compatible than a Geforce2 MX with a 440BX board. I doubt there's much difference in power draw.

I know — it’s all so strange. What’s weird is that this was a NOS box when I built it five or six years ago. Every component except the SDRAM. The PSU was a new Corsair, so it seems unlikely that it would be failing. I’ll clean the AGP slot with Deoxit and contact cleaner. Maybe swap out the CPU and DIMM.