I don't remember how many dozen tiny electrolytic caps are in that radio. We have replaced just the two or three that are bad for a customer who wants to unload the radio and keep his costs low. For him, it only needs to work long enough to demonstrate and sell. A 'scope is my go-to tool for finding a bad cap. Don't really know another way.
Replacing them all is the only way you'll have a daily driver that doesn't strand you on the side of the road a couple of times a month.
Biggest hazards of doing this are mechanical. Electrolytic caps are polarity-sensitive. Frequently the screen-print legend on the pc board will have one side of the capacitor outline marked "+". Just about every replacement cap you can buy is marked on the negative (-) side. You just have to keep track. Most replacements for the radial-lead type caps (wires parallel) will have the positive lead longer than the negative. If you insert a replacement backwards, it tends to cause trouble. And if an accidental solder "bridge" from one foil pad to the next happens when installing one of them, you'll have unpredictable fault symptoms from that until the short is cleared. Sometimes there is a bonus, where the solder bridge overloads a component and causes it to fail.
As always, taking it slow and double-checking yourself tends to pay off.
Just one problem. If the radio has any other faults in it you won't have a way to observe them until all the bad caps are replaced. Changing them all might bring the radio back to life. Might only expose the other faults that were already there all the time.
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