Thanks for your input. Well... I've tried to keep the story short, thus omitting some details that I thought wouldn't make any difference.
The board had some battery corrosion, and after removing the components in the affected area I found two broken traces. Repaired them and many others that had even a tiny bit of corrosion. Also found many of the switches didn't make proper contact. Dismantled the DIP block, it was full of green gunk. As I didn't have another on hand I restored it to factory condition. Vinegar, WD40 and a lot of work with tiny pieces 😀
After repairing the two broken traces I was very optimistic about the board, but after soldering back the components it was stuck as before, code 16. No change at all.
That's when I started tinkering with the switches, and found out a combination that allows the board to progress to code 1A. I also changed, one by one, all the jumpers on the board, to no avail.
I've tested with other KBC chips and even without one - no change; found on the net (chukaev.ru54.com) a newer BIOS image for the exact same chipset, burned it into two 27C256 but nothing changed, so this rules out a possible corrupted BIOS.
Somehow I don't think it would work with a CGA, because I never encountered so far a board that wouldn't POST without a video card (with the proper beep code / error, of course). But of course with a CGA I could use the supersoft diagnostic ROMs...
The only beep codes I get are the endless series of three beeps when I'm testing without RAM. With RAM installed, no beep whatsoever.
Here are some more pics of the board, including the battery area before and after repair.
Another quirk of the board, it doesn't seem to have a Reset header, or it is not functioning properly. One of the 4-pin headers in the lower-left corner is for speaker connection, and there are two other 4-pin headers, I suppose the Reset should be among them. But I've shorted them in any combination and the board doesn't reset. I've checked all the logic chips with my TL866 and found no problem, so this might indicate a chipset problem I guess?