It's fixed! Whitebar, you were totally right about the 33uF caps on the logic board being the problem -- despite reading correct for capacitance. I put in some Panasonic FRs and got the green light right away. I then went in and replaced the suggested PSU caps. There'd previously been a hiss coming off the PSU board; interestingly the hiss stopped when the PSU was able to successfully power the logic board. But it would come back on when the unit was turned off. The hiss stopped once the new caps were in place. Also interestingly, I don't believe the hiss was coming from any of the caps, but instead from transformer.
BTW, for the 1000uF cap, I used a 16-volt Nichicon HE, because why not. Insurance. It fits easily and costs no more, maybe less, than the 10-volt.
For desoldering on the logic board -- I actually was able to achieve this with a conventional iron. As everyone notes, it's the ground pin that's the harder one. I first got every bit of solder off the positive side, with a sucker -- till that side was unattached and loose. I then sucked all the solder I could off the ground side after getting it quite hot. After that, I reheated for as much as 20-30 seconds and wiggled the cap from the top side while keeping the pin hot. Once it was loose, I pulled it off. Board was fine, intact. For the new cap, you can simply get the ground lead very, very hot, and then let the lead draw a good-sized bead of solder down into the board. This technique is only for experienced wigglers. It would be easy to burn something, like your hand.
Thanks, Whitebar, you're a hero.