OK, I will add the resistor. The 7AF7 may be identical to the 7F8 except for Pin-out. So it might work. I'll let you know what happened when the 10K ohm resistor is added.
:: In this radio's past someone replaced the 7F8 with a 7AF7 and did a little re-wiring to make it work. I remember mentioning this here, but I don't remember when. Anyway, AM Broadcast band is working. When I got it, there was a 500 ohm resistor connecting the both cathodes together. This was incorrect. I added the correct 2.2K ohm resistor to the mixer cathode, and disconnected the 500 ohm resistor. So the BC is fine.
:: The resistor on the SW front end is missing. But I don't see how it would make any difference. It's in parallel with a coil and trimmer capacitor. I get continuity on the SW oscillator coil. Maybe this tube wont work at SW frequencies? It seems that it should. So I'm not going to re-wire for the 7F8. SW does pick up clicks and things but no actual broadcasts.
:: I haven't tried to put another radio radio near the oscillator coil to see if it's oscillating on the SW band. But I think that it's a good next step, not finding anything else wrong.
:: What should I look at next?
:If it were mine, I would put it back the way it was designed, then go from there. A resistor across a coil is for dampening; if it was made that way it was necessary for proper operation.