There were add-on cards that you could get to plug into an ISA slot that would test the hardware but it is doubtful you could scare one up now. They came with their own BIOS which took over before the one on the motherboard and boot into the test mode. You should try to identify that drive controller and the other two cards while you are at it. The middle one looks to be isa and could as easily be seated in the bottom slot. Is the top one ISA as well, the placement of the expansion slots seems odd, the middle card is an ISA slot, then a bunch of EISA and a final ISA.
Anyway That drive controller might be able to support an old CD drive on a cable with the hard drive. Then you could possibly use a floppy to boot from a CD. I don't know how that would help though, with your current problem. It might if you found an old test program that wouldn't fit on a floppy.
Sometimes you have to strip down to the motherboard to see what is really going on. There could be significant markings under the power supply or hard drive if haven't looked. Something printed on an empty space, a legend, timings or whatnot. It looks like the motherboard is fairly well annotated.