[morzh]

I [maybe not a good idea] tried a
fast power-up, and the power cord got rather warm, 

No, definitely not a good idea. It is 1)useless and 2)dangerous.

If you want to check the power transformer, you could extract the tubes (and you need to do that anyway to check them) and then power up through a know good cord and a variac and measure the transformer's outputs. Powering up a non-recapped radio is a dangerous waste of time.

Start with checking that transformer, then cleaning, then full recap, then tubes test.
Then you could try the power-up and start troubleshooting (if needed) and alignment (if it all works).