I guess the required signals don't appear at the Zorro bus to handle this sort of thing internally.
Right. They appear on the video slot but not on Zorro. But even if they would: The signal is not suitable for HDMI, the frequency is too low, and you would need at least an build-in scandoubler, or frequency upscaler.
I believe this has been tried, but the quality was not very remarkable.
With that said, I wonder if it's feasible to do something exceptionally sh*tty, like grab the RGB/CSYNC signals from Denise using an additional adaptor, feed them to the card, and process/upscale them to output via HDMI using the onboard FPGA when native video is in use?
Such a sh*tty thing already exists, namely in the form of the Indivision. All it would need is an input socket on the graphics card that connects to the Indivsion.
But yes, I agree, the unavailability of a native video pass-through is highly annoying.