My Mac Pro does not boot

The computer is this: https://everymac.com/systems/apple/mac_pro/specs/mac-pro-eight-core-2.26-early-2009-nehalem-specs.html#macspecs1

I decided to take home a Mac Pro to fix up so I could get it up and running to the maximum configuration possible. The person who owns it says he has three, but this one despite turning on regularly does not show anything on the screen. The first thing I came up with is the failed GT120, easy to replace and available for a few tens of euros. I have a few tests done with video (not live) before buying it, everything seems to be regular via DIAG led, just can't see anything. We meet halfway and I take it home. Nothing is known about this Mac, it should be exactly as it came out of the factory. The only thing is a PCI card with eSATA ports installed and expanded RAM.

I open it, I have never seen a computer in such a hygienic condition. Layers upon layers of dust on everything. Before starting it I decide to take it completely apart and clean every nook and cranny. I spend an afternoon doing everything and now the computer is perfect. Inside the heatsinks I practically found carpets of dust. In reassembling it I decide, while I'm at it, to change the CR2032 buffer battery on the logic board.

I purchased it with its original 1TB HDD. I don't know what is on it and I have no powered SATA-USB adapters to see its contents. I decide to install macOS 10.11 on a blank SSD via another Intel Mac (version supported by both 4.1 and 5.1. Not sure if it was flashed) and mount it inside one of the SATA trays.

I turn it on and as described the computer does nothing. You can't hear the sound, the fans are spinning, the GT120 fan is spinning like crazy and on the monitor you can't see anything. Basically the monitor gets a black signal, but after a few seconds it disappears showing the message "No signal." Right away I suspect that the GT120 was indeed the problem, so I decide to put the computer on the network (isolated, just it and the Mac with Remote Desktop on the network) and try to access it. The Ethernet port flashes yellow, I sigh in relief because they are passing data, but I don't see any Macs on the network from Remote Desktop. I open the router configuration and it reports no wired devices (despite the port flashing yellow). I say okay, maybe it's not booting MacOS, but at least the card with its MAC address should detect it--nothing.

Doing these tests I restart the computer several times while holding down the power button. After a few times a new problem pops up: the Mac shuts itself down a few seconds after turning on. I click the button, the LED lights up normally, the fans start spinning, and after a few seconds the CPU-related LEDs on the processor board flash and the computer shuts down. At this point I start to think it's the dry thermal pastes, since the computer from "cold" was running, while now after a while it has this problem. When I cleaned everything I did not disassemble the heatsinks from the processors (I had no thermal paste). I order thermal paste and let it sit for a day.

While waiting, I decide to do one more test. I take the GT120 (hypothetically failed) and mount it on a PC to see what happens. Surprise: it works correctly. I start to worry because the problem is starting to look more costly than it seemed.

Thermal paste arrived, I disassemble the processor board, further clean all the dust meticulously but the thermal paste does not seem exaggeratedly dry. I replace it and reassemble everything. Already at "cold" boot the Mac starts to shut down, but this time with both processors and the northbridge with the thermal paste just put on. Service manual in hand, and I start testing it com the minimal configuration.

I get to the step where the processors card is to be inserted with only slot A populated, and with one of the processors it shuts down, while with the other it does not (still remaining with the black screen). At this point I start testing the Mac with only one processor, but I can't come up with it:

1. SMC reset
2. PRAM reset: the Mac does not respond to commands. Despite the combination, the Mac does not reboot and never makes the boot sound
3. TMC reset
4. Caps lock led does not come on
5. Tried both with original HDD, SSD, and without disks (should show me the folder with the question mark) but nothing
6. DIAG leds show everything regular (photo follows)
7. No abnormal leds lit on processor board
8. RAM replaced, booted with bare minimum
9. Video card moved from PCIe1 to PCIe2

I'm searching the forums everywhere and just found some people with the same problem. Basically the rivets on the northbridge despite looking ok, would be loose and the sensor immediately goes into error. I disassemble my board, unscrew the northbridge and I can actually move it horizontally despite the fact that it is fixed with rivets.

Thanks in advance!

[Re-Titled by Moderator]