Update on the progress, or lack thereof :-(
First of all thanks for the replies and apologies for my late reaction.
So first I found that the scope has at some time in its life suffered a big impact though I'm not sure if it has anything to do with the fault or possibly something that happened after it was put aside because it didn't work. What I had originally thought small cosmetic damage on the frame left of the screen turned out to be much bigger inside with the magnesium frame being broken around the floppy drive and the frame visibly skewed, even making it hard to remove the cover. I did not manage to find any electronics actually broken though, such as cracked PCBs, components ripped from the board or anything of the sort.
I checked the power supply as per service manual and all voltages were correct and free from significant ripple or other issues, both on load and unloaded.
I tried all the dip switches that are meant to skip certain self tests. The only difference they make is that with some combination I get a different response from the diagnostic LED but I haven't managed to find out the meanings. I hate that the service manual doesn't even give a clue of what the diagnostic codes mean.
I also tried disconnecting things, trying many different combinations of thing s connected or not, just in case one of them was causing the bootup to stall. In the end I even removed such that there were only the motherboard and the power supply and still I got the exact same result on the diagnostic LED.
I then disassembled the scope completely down and cleaned everything as it was absolutely filthy, it seems mostly from storage as not much of the filth appears to be what the fan would have drawn in. I also carefully inspected all PCBs and connectors for physical damage, leaky capacitors and such. The only damage I found was a totally busted trigger chip but that is something I was half expecting even when I ordered it. In any case although this definitely means faulty triggering it shouldn't stop the scope from booting.
I then put it back together, perhaps naively hoping that by some magic it would work but as I should have reasonably expected it was exactly the same as before, except that the front power button worked and the instrument looked almost like brand new :-)
The next thing I tired was quite risky. I swapped the motherboards between my perfectly working TDS 784C and the dead TDS 694C. Now the display on the TDS 694 lit up and showed bootup in progress went through the self tests. I got an obvious failure on the acquisition board since it is a totally different one from what it expected. But other than that everything worked and it even displayed waveforms loaded from the hard disk, all the buttons and menus worked, so in short it could do everything except the most important thing. But at least that sort of pointed with certainty at the motherboard itself being at fault. In the meantime the motherboard of the TDS 694 in the 784 behaved exactly the same as when it was at home in the TDS 694 (all LEDs on, no display, no reaction to controls) further confirming the mother board faulty.
I found a board with the exact same part number on eBay and ordered it. It is from a different scope but since it has the same part number I guess it differs only in the firmware loaded into its flash. 'Only' perhaps not being the best word. In any case once it arrives I'll put it in and see what happens and how to proceed.
Thanks to Jwalling for pointing out that it is probably the NVRAM. I will order the replacements, two sets actually as the ones in my working TDS 784C are probably close to their end of life.
One thing that I have also yet to do is hook up the debugging serial to see if it reveals anything useful. I still have to figure out the connections though and make the adapter.