I'm thinking you have a problem in the amplifier that has caused that overload to the transformer and which may well have cooked it. But if you put in a new transformer, the same thing will happen.
What bothers me is that the brown wires on a Stereo 70 transformer are FILAMENT wires. That's strange unless there is a parts fault, a wiring fault, or frayed wiring, or an unintended short circuit someplace. Most failures occur in the B+ circuit (red wires, rectifier, choke and filter capacitors)
In general when this happens:
1. Especially if you do not have a means of monitoring the current the amp is pulling from the AC outlet, check the fuse and make sure it is of correct rating; not too big. Too many times, someone replaces a blown fuse with a 20 amp fuse that he dug out of the fuse box on his 1962 Ford Fairlane with a dented hood (that has been sitting behind the barn for forty years and now has mice in it). That 20 amp fuse will provide NO protection.
2. Make sure power is off, pull the output tubes and see what happens when power is turned back on. If the other tubes light up and warm up, that's an indication that there is a fault in one or more output tubes. If it pulls too much current and gets too hot, turn it off and pull the rectifier tube, then try again. If that cures it, the problem is either in the rectifier tube or the quad capacitor, or perhaps the quad cap has shorted and took the rectifier tube with it. If it still gets too hot, turn it off and pull the driver/inverter tubes, then try again. If it still gets too hot, shut it down and investigate the associated wiring in the chassis, which amounts to only what is on the circuit board and the bias circuit.
Finally, if nothing else checks out, unsolder the wires that are getting too hot and prevent them from touching each other. There is now NOTHING that should cause those wires to get too hot and burn up when the power is turned on; but if the transformer itself gets hot under this condition, it is probably shot or its internal wiring has chafed and is shorted.