You also have a PushPull output transformer that might be shorted. You can measure it from CT to each end to see if both side measure the same resistance. Also measure each of the 3 leads to ground to be sure there's no shorting to the frame of the transformer.
BTW.. that field coil should measure 800ohms.
If you remove the two 6f6 output tubes when you power on.. you should measure about +250vdc B+ on pin 3 (plate) of the 6F6 tube sockets.
C35 should have had all 3 sections replaced. Has that been done with correct polarity?
I am thinking maybe you did that as you said it died while listening to it.
http://techpreservation.dyndns.org/schematics/manufacturer.htm
Old electrolytics may appear to work "fine" while they are actually slowly leaking and getting worse.
So if these are the old original electrolytic filter caps...Replace them.
... that would be MY first likely suspect pulling down your B+ and killing the 5y3 rectifier.
If you haven't replaced these 3 caps.. do it now..
They are the worst possible offenders ....like a time-bomb in your radio.
All the paper/wax caps should be replaced as well.
These are not "optional" considerations.
ALL paper caps by now are leaking to some degree and/or have changed value.
So 1st step is to replace the electrolytic filter caps C35 - A, B & C
Then test. If B+ is ok.. then proceed.
Then replace all paper caps one at a time checking radio play after each cap change.
It is good that all caps are new!
Now, ...That coil on the speaker should, in fact, be the field coil. Remember these were the days before high quality permanent magnets.. so the field coil was placed over the pole piece in the rear of the speaker to create magnetism to return the speaker cone (attached to the voice coil positioned to float around the pole-piece )to neutral as the voice coil, with its opposing magnetic field, pulled the cone forward more or less with modulation.
If in fact the field coil is shot...
(The old speaker will not operate without the field-coil as an electrodynamic magnet)
So ...You can do any of 3 things:
1. Find another field coil or speaker w/field coil.. or get this one rewound. ( Hank Brazeal in Alabama can do it and farily priced)
2. Replace the field coil with a choke rated for about 8-10 Henries and about 100-120 ma depending on your B+ load... (may be lighter like 60-80ma)..
... you will also need a dropping resistor to compensate for the missing 800 ohms that the field coil presented.. so you can adjust the B+ properly.
3. The easiest and least costly idea ... is to remove the field coil and speaker completely and replace the speaker with a newer style 4-ohm PM (permanent magnet) speaker... and in place of the field coil you should probably use a 1k or 1.2k ohm resistor of sufficient wattage. ( this option would require you to increase the first filter cap to about 20uf and 40-50uf for the next two caps.