My first CB radio was an AM/only GE radio that I bought for a song - new - back in the day. Had to have been some forty years ago now - woof!
You can chase your tail for a long time when fixing an old radio like this. So I'd agree that replacing all of the caps can alleviate some board problems related to caps gone afoul. At least, you will know that it is not an old cap related problem in order to find the core problem - if any - which is quite possible after forty year old caps.
Couple months back, I recapped a Uniden PC-244 radio, and after testing all of the caps after the job, only three of them were still within spec or not shorted outright. Amazingly it still worked in that condition; but with MANY different issues. All of the many problems it had (even had the same problem yours has) disappeared after the old caps were booted. That radio wasn't as old as your is now. So you can see, it is due - if it is going to be a daily driver or a resto job for someone else. Finding the right bad cap may be done; but next month another bad caps creates yet another hair pulling session.
A decent 20w soldering with the right tip, a solder sucker or soldering braid to remove the old solder joints, a felt pen for marking all of the old caps in the radio on their tops, and a cheap $5 Walmart laser pointer to locate the right solder joints on the board, will all make the job go swiftly and correctly. Do it almost all the time. It will take time; but once you get into a groove you will get there w/o too much trouble.