Well, since no one responded, it seems like the only way to fix the hiss is to use the on board jumper to disable the amplifier. Which then requires rather well-powered speakers. I don't know how, but I was able to dig out something rather period-correct to pair with my heroic Socket 7 box.

This is unfortunately quite normal. There was always a competition in building mobos and cards, and price was a major factor. The moment 8-bit DAC cards appeared there was certain "standardization" of the analog section - cheap ~2W power amps with next to no power supply filtering. So it matters little what the DAC can do (and there were some noisy DACs too) if the amp picks up all mobo and PSU noises and sends that to speakers. Also back then the PSUs rarely had fan speed control and mechanical HDDs were not very quiet. So there was a background noise that drowned most of the amp noise when using cheap, small speakers.

Point is these days you just have to experiment a bit. Select a less noisy card, or better yet completly bypass the internal amplifier and use powered speakers or an external HiFi amp. Pick a less noisy mobo and PSU. The PSU should have fan control or be modded for it. And so on. You might still hear CPU noise with headphones or a good external amp, which would mean the DAC on the card is so-so, or at least it has poor power filtering. I consider some of these noises to be part of the nostalgia package, I had that back then so it doesn't bother me too much unless the hissing and such is really audible. Back in the day I was happy to have my self-made Covox play MODs, and the resistors I used to make it were not properly matched. Noise or not that was still an upgrade over PC speaker 😀