The lamps are powered off the winding of the transformer with the wire with the arrow on it (each tube heater has the same arrow as does a wire off the dial lamp section of the band switch).

What you need to chase is the open in the AC power wiring. Looking at the schematic it is most likely the main power switch or part of the mode switch/phono power switch wiring...Ohm it all out and the problem should present its self...The transformer is probably fine (though you may as well check it first to avoid wasting time on a set with an open transformer).

I'm wondering if it isn't the power cord itself because I noticed that whoever wired in the new power cord didn't do it very well as the solder job was very crappy looking.

EDIT: I figured out what it was, I had the mode switch set to phono mode which cuts the power to the radio unless the record player is turned on and since I don't currently have a record player wired up to the unit right now it was just set to off by default so when I switched it to AM or FM it powered right up.