Only GPU fan spinning when plugging in power

<< The gpu fan (radeon hd 5780) goes 100% as soon as power is connected. >>

That is NOT normal operation. no matter how hot, the GPU board fan comes on only after a press of the system-start button.

I am leaning toward Power Supply issues, not a backplane board, which has very little logic on it and is Not a common failure except maybe for overheat.

Have you checked the LEDs on the mainboard? They include several power indicators, including standby power OK. Without standby power, you can't power up.

the general technique for pursuing these issues is to remove EVERYTHING in a slot, (leaving the slide-out processor shelf in place) and reach for the CHIME. The CHIME indicates that the Processor started executing the power-on self test, and it passed. The power indicator blinking a 'help-me' code (a repeating small number of blinks) can be interpreted to gain insight.

if no joy, try also removing all the RAM DIMMs as well and try again. If working, you will get a different 'help-me' code.

The Apple Technician Guide for the MacPro 2009 (updated for the 2010) may be findable online from third-party sources. But it does NOT have deep insights about your exact issue.

it suggests that the 5V standby power led should light when the Diag button is pressed even with the Processor tray removed.

If in the US, this web site can sell you a refurbished power supply for that MacPro. (Looks like about US$50 plus tax and shipping). As always be sure you re buying the right one for your Mac. There are other suppliers as well.

https://www.dvwarehouse.com/apple/parts/power-supply/mac-pro.html

From my experience, the only parts of interest that remain in the box when you swap removable components with another working unit to check functions are the backplane board and the power supply.

You seem to be having a problem rated to power. Since a refurbished power supply is not that frighteningly expensive, and it is easier to replace than the backplane board, I recommended that. The site I mentioned also has the backplane boards available.

experience with these units tells us that when they unit spontaneously power on with the power cord, it is VERY likely a defective power supply. (The fans are running off the 12 volts.). There are no high-power relays on the backplane board. Typical Apple power supplies only supply a single signal to the power supply, that says GO.