I have a diamond riva 128 PCI happily running in a K6 system. For a card that old it is indeed a rather unusual behavior.

Define work. I had a 128 in the day that would blank out in dos and only show an image in windows.

With the card inserted there is no picture and no activity from the system - systems are all DOS based.
With the P35 board the card shows it's signature and proceeds operation accordingly.

Mine was in an s7 and developed the flaw after a transient power cycle.

If it were mine, I'd inspect the card carefully with a magnifying glass for any imperfections or bits of debris, brush the surface of the card with a dry tooth brush then use a pencil eraser to shine up the contacts.

It has to have some flaw that is only a problem for older systems.

Ok, so I'll see if inspection/cleaning yields something. If it is like in vvbee's case then I'm stuck with it only working in a more capable system.
I suspect that(surreal hypothesis ahead) some minor damage occurred and thus only a more "durable" PCI interface can feed/stomach the card which is found on, for example, the P35.

Try rubbing the contacts with an eraser. It brought back a "dead" Voodoo 5 of mine once 😀.

If it doesn´t show a picture in DOS but works in windows there may be some BIOS corruption.
You could try a reflash.

As mentioned before it should work with older systems. It was the first PCI video card I ever owned, switching from a ISA based ET4000 on my old P133.
Board was a Gigabyte GA586VX if I remember correctly.
Yes, I really used an ISA card in a PCI board for a while ´cause I couldn´t afford a PCI graphics card as a student.