Inconsistent 2 or 4 LED; sometimes Chimes; sometimes even works!

EMC 2546 with Fusion Drive. Was working fine, then...inconsistent problems.

Status: I've pizza wheeled apart the screen to see the LEDs. The only USB device connected is a basic Dell keyboard, known good.

Symptoms:

• Always responds to power on button. Drive spins up, fans on every time.
• Sometimes (perhaps 5 out of 10 attempts) powers on with 2 LEDs only. No chime, no backlight. USB non responsive on keyboard caps lock light.
• Sometimes (perhaps 3 out of 10 attempts) powers on with 4 LEDs in sequence and evenly lit. Backlight comes on but no display, no chime. This happens most often when NO usb accessories are plugged in.
• Sometimes (perhaps 1 out of 10 attempts) powers on with 4 LEDs in sequence and evenly lit. Chimes and backlight comes on. White screen. No further response; will sit on blank white screen for multiple minutes. This happens most often when NO usb accessories are plugged in. When plugging in a keyboard at the white screen, caps lock is unresponsive.
• Sometimes (perhaps 1 out of 20 attempts) -- I get 4 LEDs and a chime with a white screen, but this chime will replay quietly and about half normal length (so perhaps half a second) in a loop every few seconds. This continues. This happens most often when NO usb accessories are plugged in.
• Sometimes (perhaps 1 out of 20 attempts) -- everything works. Chimes, LEDs, boots and runs A-ok until the next power off or restart. No apparent issue whatsoever. This happens most often when NO usb accessories are plugged in.
• When disconnecting the screen and powering up, 3 LEDs typically come on and fan spins up to high speed. No chime.

Tests:

• Have attempted swapping with another USB keyboard.
• Swapped power supply cable, used a different wall outlet.
• Swapped known good single RAM stick in from another machine.
• Disconnected (and reconnected) webcam.
• Disconnected (and reconnected) SATA hard disk drive.
• Reset PRAM (when I can get the keyboard to work with a 4 LED boot and before the typical white screen freeze) -- successful double chime.
• Reset SMC, have left it to sit unpowered for a few days too.

I've inspected the logic board, screen, PSU, and and so on from the side accessible with the screen off. There is no obvious damage, liquid spill, burn/short markings, etc. Everything seems snug. I've disconnected and reconnected what I have access to on the facing side. I've also cleaned (99% iso) and inspected the USB ports.

The only thing I can do that seems to make any difference at all to the random start states is NOT plug in any USB accessories. I don't believe it's ever reached a white screen (much less booting properly) with any USB accessory plugged in since this issue began. Though, I do plug in keyboard/mouse without issue once it's booting properly (when that rarely happens).

Repair?

The inconsistent behavior but normal operation on the (rare) proper booting makes me think it's the PSU. But, the strange relationship with USB accessories makes me think it's the Logic Board. As I don't have a donor, I'll have to order one part or the other and would like to get it right on the first try if there's an obvious conclusion here. I've got a multimeter and basic skills to test voltage or resistance and can find a boardview if that's helpful.

Thanks in advance for any advice. I'd love to get this machine back up and running.