I notice in all your pictures there's no BIOS battery installed. Does the board behave the same if you add the battery?

It is worth to reseat all the socketed parts, sometimes it makes a difference. Two of the boards have an OTP EPROM and those shouldn't suffer from bitrot right now at all, nor the UV one since the orginal label is still intact.
Is the CPU jumpered right ? It might be running out of spec. There's garbage on the screen which suggests data getting corrupt along the way and that suggests too high ISA bus speed or perhaps dirty connectors which could cause issues since BIOS usually sits on the ISA bus.

All right, problem solved, all of this was caused by EDO RAM modules... i was aware that 4DPS requires FPM memory and I thought that I was actually using FPM. I tried many different modules and the result was the same. But it turned out that all of them were EDO and I stored FPM in different place 😜

So this is how Tomato Board behaves with EDO memory.

In the process I also thought that my pico PSU could cause the problem, because it lacked -5V. So i modded it with LM2662 module (+ 10uF ceramic cap and Schottky diode). Now I know that it wasn't necessary, but maybe someone will find this idea useful.

BTW: battery is also unrelated to the problem.