My guess is that if the caps were drawing enough leakage current to present a significant extra load on the bridge (an amp or so, maybe?), they would be vented more violently than the mild bulging we can see. That much heat burned up in the caps would be obvious with shrinkwrap melted off, etc.
No doubt the caps are bad. Probably sky-high ESR. Typical Chinese capacitor failure stuff. Outgassed enough over the years to bulge the can, and vent pressure, drying out the cap. I don't think it really contributed to the catastrophic failure here, though.
What is that melted-looking yellow wire connected to? Does it look melted from excessive current through it, or was it melted by touching that hot bridge?