there is a 32.768 kHz quartz oscillator on the left of the topmost 16 bit ISA slot. small metal barrel. You can try replacing it, they sometimes fail. Should not be hard at all.

Good call. I've ordered some from Ebay. 😀

There is a similar motherboard at
http://th99.classic-computing.de/src/m/C-D/34280.htm

It shows a jumper (labelled W14) on the edge of the board near the processor for
"battery type select internal" as closed
and "battery type select external" as open.
I can't tell from your picture if that jumper is there on your board, or whether it's open or closed.

My board is a different revision and that jumper is in a different place. You can see it just below and to the right of the keyboard connector. The image is old, from before I mounted the external battery. I have the jumper open in reality, and the board saves the settings just fine, so the external battery works.

So, an update on this issue:

Turns out my hacked together 3V external CR2032 battery was the culprit. I guess the 3V wasn't enough to keep everything working when the computer was turned off, and it really needed 3.6V like the internal battery used to supply.

I soldered on a new Ebay-bought 3.6V Ni-Cd barrel battery to the motherboard and now everything works fine, including the RTC. 😀

Incidentally, this was my first "serious" soldering job in 15 years or so. Shows you how experienced I am at soldering. 🤣

Apparently, there are double stack CR2032 holders which will hold two CR2032 batteries in series. What I did was to use a single CR2032 holder, the type with a lever arm, like a hair clip, and bent it so that it can accommodate two CR2032 batteries. I never wanted to use those barrell batteries again.

Of course, you need to remove the trace which tries to charge the non-chargeable CR2032 battery, it usually involves cutting a through hole diode lead.

If you think the 6V is too much, you can add a diode or two to drop the voltage some. I've seen 386-era boards require voltages from 3.x to 6.x volts for the battery.

Or you can do what I did. Buy yourself a barrel battery, but do NOT solder it back on the mainboard. Instead just use CD audio cable and solder it on the battery as extension and plug the other end to the battery socket. Next, encapsulate the battery in small plastic container and stick it on the case where no electronics could be possibly harmed, usually in the front of the case under the speaker. If it leaks, it has a looooong path to travel through the cable, so that gives you quite a long time to notice and replace that bad boy. 😉