I had experienced similar problem in the past, except in my case, FDD would not initiate disk spin, but it would do a seek test while POSTing every time. This problem was caused by a crappy FDD cable, I guess it was internally broken somewhere.

Does the /INDEX signal from the drive reach the controller?

It could be damaged DMA2. I've read about it. But first of all it's better to disable all caches and to try again.
I suppose FDD bios setting is correct, jumper settings are correct, the FDD is working.

/INDEX reaches the controller as do all other signals, this is not a problem with connectivity or drives. The behaviour is same even with an IO card, drive+cable+controller that works in another computer does the same stuff as the integrated controller on this machine. IRQ6 does show activity on the scope aswell, it seems that it isn't reaching the CPU or something, it definitely reaches the chipset.

Turning on/off L1 and/or L2 cache has no effect either. Floppy type etc. settings also play no role.

I have traced around a little and IRQ6 is only connected to ISA, SuperIO chip and VT82C486A chip. Unfortunately no datasheet of the VIA part exists in the wild it seems. I'm more or less confident now that the problem is in that chip... It seems utsource has it for sale for the amount I got refunded for lack of VLB slots (I have soldered back some now) on that board when I got it (15usd 🤣).