[Phlogiston]

The sweet smell is probably the bee's wax used in the transformer.

If the transformer is already damaged, other than rewinding it, there is nothing you can do to fix it.

If it runs without getting warm for 15 minutes, it is a tough call. Most with shorted windings will begin to smoke/bubble by that time. Still it could have a few shorted windings in the HV secondary.

With the rectifier removed and the radio unplugged, measure the resistance from the AC input pins on the rectifier socket (2 of them) to the chassis ground. It should be nearly equal. If there is a difference of more than 5 - 10% one side of the HV winding probably has shorted. The readings will be in hundreds of ohms. This can be done to the other windings as well but is far less likely to find a problem since they are going to be in the 10 ohms or less range.

Lastly run the transformer as you have SUPERVISED, for an extended time - with the rectifier removed. If it is shorted it will eventually smoke. Don't let it continue. Don't attempt this unless you have the time to watch it carefully. Don't want a smelly mess or worse. If it fails, it was going to fail anyway. Look for a new one BUT (!) check out the radio thoroughly before installing a new one. Many people ignore every component except the caps. a bad resistor can cause excess current to flow in an output tube - enough to cause power transformer failure.

Lastly, don't do any of this unless you understand the dangers involved with servicing high voltage equipment.