I've found this is a recurring and extremely common problem in iMac G5 iSight systems. Everyone always says "capacitors!" when talking about G5's, but the caps in these machines are almost always nice and flat.
The issue is indeed in the GPU. In order to repair these machines, I've taken a similar approach to other BGA soldering related problems and gone with a heat gun solution. Oven/toaster/etc reflow will NOT work here due to the through hole electrolytic caps on the board which can NOT take solder reflow temp. Here are the steps I take:
1: Remove the logic board from the machine. iFixit of course has excellent instructions.
2: Remove the two small heatsink assemblies from the front and back of the board. They are held in by four spring-loaded plastic clips which can be released with a pair of needle nose pliers. Be careful! If you break the clips replacements are impossible to find, and those heatsinks are necessary.
3: Fold 3-4 layers of heavy-duty aluminum foil. Use this folded shield to cover the back of the board (the side where the power connector is). Fold around edges and capacitors. You do NOT want to heat the caps unless you want them to pop, which then requires more work to replace them. Press the foil down well and you can see the GPU create a nice intent in it. Then take a small hobby knife and cut away the foil ONLY around the GPU, giving maybe half an inch clearance on the sides.
4: Place a small piece of roll solder on the GPU, maybe 2mm. Make sure it sticks with a small dot of thermal grease. I use this as a temperature indicator . . . . a laser thermometer works too, but I find this method is quite foolproof.
5: Heat, SLOWLY, with a heat gun on its lowest setting. You do NOT want it blowing so hard it is going to fry things or blast away tiny components. Mine is a two-speed $10 model from Harbor Freight, and it's always served me well. Hold first 3-4 inches away for at least a minute to preheat the surrounding board and chips evenly, then close to about an inch away to really heat up the GPU. Use a circular, even motion and be sure to heat the whole chip package evenly. When the solder you put there earlier becomes a nice pretty shiny molten ball KEEP heating for an additional 60 seconds or so (your solder is leaded unless you're buying from some odd source, and the GPU's is lead-free which melts at a slightly higher temperature).
6: Turn off your heat gun and STEP AWAY. Do NOT jolt or otherwise molest the board! Allow it to cool for a minimum of 30 minutes, then make sure it is COMPLETELY cool to touch. Only then should you remove the tinfoil. Closely inspect the surrounding capacitors and insure that NONE have popped or developed a domed top from the heat. They should all be perfectly flat at the top. If not they will need to be replaced as well, but if the tin foil was positioned properly and you didn't wave the heat gun too much none should be.
6: Thoroughly clean the GPU surface and that of the large northbridge chip on the other side of the board, do the same with the heatsinks. Remove all the old thermal material. I use Artic Silver's Arcticlean two-part cleaner and I highly recommend it, but 91% isopropyl alcohol will work in a pinch with a bit more labor. Then reapply thermal grease to the GPU and northbridge chips in a thin, even layer. USE THE GOOD STUFF. Arctic Silver 5 is my preference, but AS3 is okay too. Poor quality heatsink grease and uneven application of it is a large reason these problems happened in the first place!
7: Reattach the heatsinks. Be mindful of the small piece of tape on the GPU heatsink, it goes where the two power handling chips are on the top left side of the GPU. Else the metal heatsink might short out the SMD capacitors located there. Reassemble according to iFixit's manuals for Logic Board replacement.
I've repaired about a dozen iMac G5 iSight machines in this manner. It has worked, without fail, EVERY TIME. I have yet to have a customer return one.