Yup, bad caps, and they'll give that exact kind of symptom. Lots of things from about 8-10 years ago had bad caps. The way I heard the story, a name-brand capacitor maker leaked a fake recipe for a key ingredient, which all the chinese suppliers then copied. And a couple years later, everyone had failing computer monitors, motherboards, power supplies, TVs, you name it. For a while, I had a side business repairing computer monitors with blown caps.
The goop all over them is probably just glue to hold things in place and avoid metal fatigue from the vibration. Replacing those caps is actually quite a simple repair--10 minutes with a soldering iron and $0.50 worth of parts. With all that available board space, you could even upgrade that power supply to get a smoother output. Heck, pay for shipping and I'll fix it for you, if you want. PM if interested.
I'd suggest you look around the board to see if any other capacitors are bulging or leaking, or are of the same brand--you'll want to replace any suspect parts while you're in there.