First, anything is repairable. Secondly, there's nothing particularly magic about electronic circuitry.... it just takes time and patience to inspect, troubleshoot, and verify components. I agree that the solder work has definitely been retouched since original manufacture, and pretty hamfistedly, and maybe more than once. Clean up the boards with isopropyl alcohol (preferably 91%, not typical rubbing alcohol) as already suggested. You're discounting mechanical problems too early. Maybe a weak ground is causing a component cap to cycle and discharge causing the pop, or a voltage regulator to swing when it should remain stable...I have no idea really, other than I know it isn't as simple as "bad mechanical connections only cause random intermittent events." Any capacitor that's almost 50 years old can be suspect. I've done complete preamp and power amp circuit board repopulations on Dynaco integrated amps, rather than try and figure out which pesky cap, resistor, diode, or whatever is causing a problem. It's relatively cheap to do, and just takes time.
If you don't want to mess with it, any competent audio repair facility should be able to troubleshoot and repair your circuit for you.