I've seen weird things happen with computers, that in many cases I thought was a power supply issue, when it turned out to be some completely different other component. One time, I saw a computer that wouldn't boot until disconnected the power from the CD-ROM drive, but left the IDE cable connected (it would fail to boot if you removed the cable!); on that one, it turned out to be a faulty network card (don't ask me how - that's just the way it turned out).
In order to really figure out a problem like this, you need to have on hand known-good components, and swap out one at a time while observing the results. The only problem with this scheme is if something is smoking parts - you might plug in a know-good component, only to have it turn into a dead component.
TBH, from your description, it sounds like it is likely a bad (or going bad) video card; have you noticed any other video related issues (things you thought may be due to the monitor - and not the video card)? Things like fuzzy graphics, noisy output, changes in color or brightness at random times? In many cases, you'll find these issues are due to bad caps on the video card.
Regardless, it isn't worth replacing caps anywhere on a computer system - it isn't worth the trouble, time, or money - unless you have absolutely no choice in the matter, of course. It is far easier and ultimately less expensive to just purchase something to replace the part and move on (throw the old device in your junk pile).