Yes.
If you/we see something that we think looks odd, then that's when you might have to remove something to test it. Once done checking everything and if nothing pops up, then it might be time to start pulling components and checking out of circuit.
Yup, alligator clips will work for that. Or even better, as Per suggested, connect the series bulb there.

Which is why I still do it myself, even though I've never had any accidents with that before. However, I did have improper wiring accidents, and the bulb did save me there are few times. Hence why I still think it's worth doing.
Looks like leftover flux from the factory soldering... but then it's also odd that this would be the only spot to have it while the rest of the board looks pretty clean.
Try cleaning it with some IPA and see if that changes anything. Probably not, but you never know.

R27 (100 Ohms) on the secondary side looks like it ran a bit hot. Probably OK, but check it anyways.
R3 and R4 (both 47 KOhms) also look like they ran hot before. Worth checking those too. But again, make sure the primary caps are discharged (less than 1-2V) before doing that. Otherwise, you may damage your meter.
Finally, the new cap you soldered for C8, while not a problem for the time being (for testing), you may want to re-solder it as close to the PCB as possible once you fix the PSU. Vibrations from the speakers can cause joints to crack over time. (Hence why some components are glued down.)
Speaking of glue, I do see a bit of the problematic tan glue on the primary side of your PSU. But it doesn't seem to have turned dark brown so it's unlikely to be the cause of the issue. Otherwise, this glue is quite often a problem in electronics, particularly in places where things run hot. The heat bakes it into a dark brown color and then it picks up moisture from the air and becomes slightly conductive (in the high-KOhm to low MOhms range, which isn't a lot but could still disrupt some circuits.) So at this point, you can just leave it as-is. But once the PSU is fixed, you may want to remove it from components leads and exposed metal parts.
Yes, the slow start-up definitely does sound like a capacitor issue.
As electrolytic capacitors heat up, their electrolyte becomes more reactive, making the capacitors' ESR become lower.
A quick way to test a device that you suspect has dead or dying caps is to blow some hot air on the PCB with a hair dryer until everything gets fairly warm (but not so hot that you can't touch it.) If it works normally right away afterwards, there is a high chance capacitors are the issue. Of course, in some cases, it could also be bad/cold solder joints too. However, these are usually a little less responsive to heat changes and more to vibration / movement of the device.
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