Joe,
The large cap in the power supply is a filter capacitor and it's doubtful that it has anything to do with your motorboating. All it does is filter out the ripples in the supply rails. Does it do this on both AC and DC? If so, then for sure it has nothing to do with that one.
You have not wasted your efforts. Regardless of the issue, the recapping will definitely improve the sound performance and future reliability.
The electrolytic caps in the tuner can be changed if you like. They dry out and deteriorate just like all the other caps in your boombox. Fortunately, they usually do not affect tuner alignment since they are typically used as bypass caps or audio coupling caps. Keep in mind that older caps like that have as much as 20%+ tolerance. Any capacitor critical for tuner alignment needs to be greater precision variety. However, unless the problem only occurs during tuner operation, once again, it's likely not related. There are critical caps that you do not want to change in the tuner, but those are the ceramic and styrol and other similar caps. They would likely affect alignment or the oscillators if replaced. Another reason to avoid the tuner is that while handling the board in that location, it's very easy to distort the tuning coils. This can happen even if you are careful and mindful -- it's very easy to do since it's very hard to keep track of all 10 of your fingers during handling.
The corrossion issue is another matter. Motorboating can be related to bad caps but there is a relationship with the related resistance/capacitance in the circuit, either of which can shift. Because value shift is much more prevalent with aged electrolytic capacitors than the resistors, they are generally suspect before resistors are. Many PC boards back in the day had early double-sided designs, which are not very reliable. Todays boards have copper on both sides (and sometimes in between in layers too). Early boards had copper on one side and printed resistors or traces on the flip side. The "printed" conductors are notorious for failing or shifting. If I see corossion like that, especially in the areas where the top traces are, I'm whipping out my Fluke meter and needle probes to check for continuity and resistance. In fact, what I do is find the origin of a trace and follow it top/below and see if continuity is intact. Don't do short segmented checks unless your check on long bottom/top/botton traces shows an issue. Then you can try to narrow where the issue is. For example, you can check A-B, B-C, C-D, D-E, E-F. But for me, I'd much rather just check A-F and if continuity is good, then there's no point to check C-D, right? I would just use a horsehair applicator brush with some isopropyl to clean the green stuff. You can follow up with clear water if you like.
You also never mentioned detailed testing. Too often, I read "my boombox does this... what's the problem?" For a better analysis, it's much better if I read "The sound is like THIS in A mode but not B; It sounds bad through speakers and headphones but outputs a proper sound via line-out; it only occurs on A side; it happens on both AC and DC, etc. etc." Armed with these clues, we can certainly make a better analysis as these symptoms and clues helps to eliminate certain things (mean more time saved, and less wasted work). Do a complete and through check, even if you don't think it's related.
Last thing.... on at least 1 picture, I see new cap leads poorly soldered (or perhaps they were not completed yet). I presume you photographed it and touched up those pads after? If not, then you definitely want to check and ensure that all joints are soldered well.
Hope this helps.