Should the A1200 output a video if I power up just the motherboard with no disks attached?
Yes. The amiga should boot to the instert disk screen without both floppy drive and hard drive and i suggest that this is what you do. I believe the led blinking pattern is a error code, but i sadly do not know the error codes myself. I am sure there are plenty of people here who do though?
I dont know if you are good with voltage meters and such? Because it could just as easily be the PSU that is getting old and is not outputting the right voltage anymore. So checking this with a voltage meter might be a good idea.
Also if you try it without both HD and floppy, then also make sure you also disconnect power cables as well as the ide cables.
The software failure can be either due to buggy software or a hardware problem.