Okay. I did a couple of tests to rule out that capacitor. First, I removed it, tested it with my fluke. It tested almost right at 1000uf. I also tried two separate 1000uf capacitors rated at 16V that I have on hand. It did not fix the problem. I switched the two capacitors, and the problem persisted.
If you swapped the L/R output capacitors and it made no difference, then the problem that you are having probably isn't the capacitor, although if it has leakage, I would change them anyway. Something happened to it whether it was scorched or rusted or leaked, I wouldn't reinstall in that condition. It's cheap enough. That being said, you keep mentioning that you "checked" the capacitors, however a capacitance test is only one thing that goes bad on a cap. There are other problems too which are even more important which is leakage and ESR. Not sure if your fluke can measure those but I think not. You will probably want to get an LCR meter for that, although in practical sense, it's far easier and simpler to just change caps.
No disrespect intended @SupreDuper, sorry, my mistake.
@HRmeteohub: None taken -- I just wanted to clarify why I stated transistor and not IC, quite simply because that is in fact a power transistor and it's function is most definitely voltage regulation. The rusty portion is nothing to do with the transistor -- it's the clamp on heat sink.
The problem is in the signal path. One of the first things I did was switch the speaker channels to see if the problem switched sides. It didn't. I also checked with headphones it did not circumvent the problem.
Tonight, I checked all of the electrolytic capacitors in the stereo signal path. They all checked fine. I went ahead and replaced all of the capacitors that were 1uf and smaller with metal film capacitors.
The problem is still there. I'm going to order a replacement LA4500 off eBay. I'll let everyone know if that fixes the problem.
Again, with the capacitors, check ESR and leakage, not capacitance but you already changed them so hopefully they are out of the equation. As for the amp chip, it may indeed be bad, as amp chips do go bad frequently enough and in the beginning, it does seem like they start out distorting as they fail. However, the simplest thing to have done in the first place instead of continuing to just throw parts at the boombox and hope something fixes it, would have been to take a logical approach AND START WITH A SCHEMATIC DIAGRAM. At this point, if I were you, I would desolder the L signal input pin of the amp IC to divorce it from circuit, then short the L/R input pins to bring the (clean) right signal to the L side as well and see if you now get clean audio out of amp. IF you are certain that the headphone jack and EXT speaker jacks are both clean and passing good signals, then you can reasonably conclude that:
IF issue fixed, then AMP chip is not the source of the problem and the issue lies before the amp chip.
IF the issue persists, then the AMP chip is probably bad.
On the other hand, this approach does require the schematic so you might as well just try a new amp chip and see if the problem is fixed.