Those wires are not burnt because if they were, the wire along its entire length would be melted. Your hum is caused by an open filter capacitor not a shorted capacitor. Since you only turned it on long enough to hear a hum come up, you probably noticed that there was no smoke or a burned fuse. A shorted capacitor would blow a fuse, smoke, or both.
At least it is not shorted but is still bad and needs to be replaced. Those one's I circled are the one's I would start with. It looks like one of those has 4 individual capacitors inside, and the other one looks like it has 3 capacitors inside so for a total of 7 new individual capacitors or 2 replacement cans. Get these replaced and I'll bet you my left thumb that it will work and work well. After this, then maybe think about the other capacitors and maybe any out-of-spec resistors. Transformers are the least of your worries at the present.
Those old orange paper can capacitors are notoriously well known to not hold up well after a few years. Today's capacitors will outlive you and me!