[Raleigh]

Pin 5 is the grid of the oscillator section of the 6A8. The RF from the antenna goes to the grid cap. What kind of signal do you see on pin 5?

I'd guess the most likely culprit is either the band switch or one of the push button switches. If none of the push buttons work, I'd suspect the band switch. You could clean the switches with contact cleaner before you do further tests, just to see.

A simple test will determine if your local oscillator is working when in push-button mode. You can do this easily with another radio placed very near your 39-30. Turn both radios on. First, experiment with the 39-30 set to dial tuning on broadcast band. Tune the test radio to a station high up on the dial, say around 1500. Your IF is 470 KHz, so then tune the 39-30 470 KHz lower than the test radio, to around 1030 KHz. The test radio should begin to squeal. (It also might squeal at other frequencies, even when the two radios are at the same frequency. Depends on the test radio, but any squeal of the test radio means your 39-30 local oscillator is working.)

Then try the same thing using one of the push buttons. Since you don't know the frequency the push button oscillator is tuned for, this time tune the test radio up and down the dial until the test radio either squeals or it doesn't. Then check each push button in turn. If no squeals then the local oscillator of the 39-30 isn't working in push button mode.

If no squeals, check continuity of the push button coils. That should be easy. Then check continuity from pin 5 of the 6A8 to the high side of resistor 9. Resistance should be about the same as the resistance of the coils. If no continuity or much higher resistance, then the band switch or the push button switches are dirty or bad. Also check resistor #9. I don't think the push buttons will oscillate without #9 being somewhere near its rated 51K ohms.

If your LO is working but you still can't tune any stations with push buttons, then check continuity from the grid cap to the high side of the antenna trimmers with the push buttons pushed in. There shouldn't be much resistance. If high resistance, then the push button switch or the band switch is dirty or bad.

If you have good continuity through the switches, then suspect those 370 pf micas. There's not much else in the push-button circuit to go bad, if I'm reading the schematic right. (I might not be.)

P.S. You might check the continuity of the antenna coil, but I THINK if both BC and SW bands work, then all parts of the antenna coil are good.