MacPro early 2009 (4.1) can't/doesn't want to boot

This Mac is special — I have a Mac Pro Early 2009 with dual CPUs. It boots, but only when it feels like it.

A few months ago, I figured out why it wasn’t starting: a component was shorting everything. After removing that part, it finally booted.

Yesterday, I tried to boot it again to install the latest supported macOS (El Capitan). I managed to update one of the systems, but the second one wouldn’t boot (i have two MacPro). To troubleshoot, I disassembled the machine and tested the parts outside the case, plugging the main board into the processor board with the PSU. Nothing happened.

Feeling disappointed, I reassembled everything back into the case — and it booted! But now, once again, it refuses to boot. I don’t want to keep disassembling and reassembling it, hoping it will eventually work.

What worries me most is the PSU: it makes a whistling or high-pitched noise, almost like it’s charging up to handle the boot sequence. It seems odd that the PSU could be failing, since yesterday I was able to restart the Mac multiple times without issue.

So I need help deciding: should I just buy a replacement PSU on eBay for about €100 (~$117–120), or should I keep investigating other possible causes?

For context, here’s what I’ve already tried:

• Resetting the BIOS, PGOOD, SYS, and SMC using the pins on the main board
• Reseating each RAM stick
• Reseating the two CPUs on the processor board
• Reconnecting every cable

I’d appreciate any advice on what to do next.