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It does sound like a motherboard issue given the steps you've tried out and the PSU BIST is passing. That said, I would bring the unit down to minimum to post to see if you get any other signs of life. So that would entail fully removing the GPU, SSD, and RAM. With the parts removed, you'll then want to turn it back on to see if the power LED is flashing any diagnostic codes. A responsive motherboard should be 2 flashes at the power LED since it'll have no memory. If you get that, then I'd reinstall the parts one at a time to see what may be causing a short, if any. If you don't get that memory error, then motherboard would seem to be the way to go. 

I agree with @DELL-Daniel V  and the "simplification" of the system for troubleshooting, however, I don't usually go to those extremes.

Have your tried only connecting wired USB keyboard and mouse, and then also the monitor. Remove any USB dongles and any optional Add-In PCIe cards.

Does that system have on-board video you never use? If so, now would be a good time to remove the video-card and try to use it.

Remember to always dissipate the flea-power after it's been on, and you are getting inside again.

A PSU on its way out can still spin the fans with the test button pushed.  Does it turn on if you press PSU and power buttons at the same time?

I agree with both Tesla and Daniel for troubleshooting.  Use Tesla's simpler method first, then Daniel's second.

Kind of rare MB goes out.  I'm leaning toward the PSU.

@bradthetechnut​ ,

1. A PSU on its way out can still spin the fans 

2. Kind of rare MB goes out.  

1. AFAIK, the "Test button" is just a Dell thing and I can never really remember what the dead machine is suppose to do when you press it. 

When the need arises, I just test my PC-Power Supplies (SMPS) with a Digital PC-Power-Supply Tester.

2. Agreed (unless there was the occurrence of an "unusual event").

@Tesla1856​ 

Our PSU's have 1 of 2 possible built in self tests. There is a button BIST and a buttonless BIST. In this instance with the R12, we should have a button BIST. Here is one of our articles outlining the steps and results of testing the PSU with each version:

Power Supply Unit Built-in Self-Test (PSU BIST)

The article says its for our newer Dell pro models but really it applies to all of our PSUs!