Now that I've fixed its broken solder joint, and I know the card works
in linux, I can verify the DOS flasher DOES NOT work. It gets far enough
to trash the chip's BIOS and then fails, replacing the card with a brick
that hangs the computer at boot.

I doubt it works with the Windows flasher either. Other sil3114 cards
may flash, but not this one. It has a r5403 BIOS soldered on. Take it or
leave it.

I found the Lattice archive for legacy SIL cards:

https://www.latticesemi.com/Support/ASSPSoftwareArchive

There you can find a 3512 BIOS, version 4.3.70. It seems unique, they
call it an "IDE" BIOS. I flashed it onto my 3114 card, and it boots.

It also works in DOS. I tried two later versions of the 3512 BIOS. Both
boot, but like the 3114 BIOS, fail on a file copy. So there's something
special about the 4.3.70 BIOS that lets it work in DOS.

They call it an "IDE" BIOS. Maybe that means IDE vs AHCI. Don't know.

Though it worked with a DOS file copy, I could not get it to boot from
the SATA drive, in the computer I had in mind for it. Maybe the BIOS is
too old and the hard drive is too big. Or maybe it's incompatible with
the motherboard BIOS. Don't know.

Looks like they gave up supporting DOS after that version. The archive
has no such BIOS for a 3114 card.