Static heard in the speaker when fooling with the volume is a dirty volume potentiometer. You electrical cleaner to fix that, spray it in the pot, cycle the knob back and forth. If that doesn't sort it out, time for a new potentiometer.

As for no audio, well to ask the obvious, do you have an AM station around you?

I would use a signal generator, am modulated at 11khz, as the source and trouble shoot it that way But I have those items and an O'scope to signal trace. Might be something as simple as just needing an alignment. Hard to say without those tools adn a schematic.

Isn't that OEM radio a tube type radio?

Would love to see a photo of the inside

Good memory Cab!

It is likely that 1964 Plymouth radio still has psome Tubes inside.

from

https://www.caranddriver.com/features/a15128476/the-history-of-car-radios/

1963: First All-Transistor Radio

A number of manufacturers introduced transistors to their aftermarket car radios in the early 1960s,
but (German manufacturer) Becker’s Monte Carlo model was the first to be fully “solid state”—no vacuum tubes.

I guess that also means finding an antique radio repair shop, and the chance that replacement tubes may not be available.

but Wikipedia says:

In April 1955, the Chrysler Corporation announced that it was offering a Mopar model 914HR branded Philco all transistor car radio,[11] as a $150 option for its 1956 Chrysler and Imperial car models. Chrysler Corporation had decided to discontinue its all transistor car radio option at the end of 1956, due to it being too expensive, and replaced it with a cheaper hybrid (transistors and low voltage vacuum tubes) car radio for its new 1957 car models.[12]
In 1963, Becker introduced the Monte Carlo, a tubeless solid state radio with no vacuum tubes.[10]

https://en.wikipedia.org/wiki/Vehicle_audio

Thanks for all the advice...I'm sidetracked right now, hopefully I can get back to sorting it out soon. Keep the advice / tips coming!