I've no real experience of dealing with knackered Amigas... but I've been fixing a lot of Sinclair Spectrums of late so I guess it's the same principles.
What I've done on the Spectrum is make my own diagnosis ROM which tells me what works and what doesn't on the motherboard. I don't see why you shouldn't be able to do the same thing on an Amiga.
The first thing I would do is burn a simple ROM with something like (pardon my asm, it's been a while):
START MOVE.L #$0000,D0
LOOP MOVE.L ($DFF180),D0
INC.L D0
JMP LOOP
Now from memory, and it's been a long time so I've probably got the syntax/numbers wrong, but this would just increment the background colour, resulting in multicoloured strips all over the screen - if the system is working. If this didn't happen, I'd know either the display was dead chip or the CPU was dead or not getting a clock signal. From there you can build more features into the ROM to test various parts of the system.
Again, I'm probably way off on the details, but I think the principle should be sound.