Excellent progress! You have something coming out of the speakers and the filaments are not burning out. So you know you have B+, proper filament voltages and a working output stage.
That 100pf cap might well have been shorted. You can check them with an ohm meter by applying and reversing the leads. When you reverse the meter will deflect from the discharge. It's not a great test since it does not test the cap at working voltages
Here is a link discussing capacitor testing
Your earlier problems might have been from the caps forming and resistors can change values. You can check them. It might be also that 125 volts just stresses the set too much.
On really old sets I just replace the electrolytics and the wax covered caps. Makes life easier.
As far as reception, if someone had been fiddling with the set all the transformer cores and trimmers in the IF and tuning might have been changed.
After dark you should be able to pickup stations across the country. But there could be alignment problems or some other cleaning issues in the tuning section. Typically alignment does not need more than a tweak if anything. But if someone was messing with the set who knows?
So the next question is do you have a signal generator?
Here is a link to an alignment guide for AA5 radios Yours is not an AA5 but and you have the procedure on the schematic (assuming you have rf generator). But he describes how to use a 2nd AM radio for alignment.
BTW WA2ISE posts here as well and has written some very interesting and informative articles online.