Optiplex 7040 MT Now Cycles On and Off but Never Boots

I bought this Optiplex 7040 MT used, and I have been using it for a few weeks. I have Windows installed on a 2.5 in SSD and then I'm using a 16TB 3.5 in HDD for NAS-like storage.

The one problem I was having is that there was a lot of fan noise, and a whining, like perhaps something with one of the fans was not working properly. I tried disconnecting the two things that look like fan headers in the bottom left of the motherboard (ie near the I/O panel, about even with the HDMI port) and that is when the trouble began. I moved the computer out of where I was using it so that I could open the case up and unplug these two headers on the motherboard. I then plugged it in to a monitor with just a mouse and keyboard plugged in, with the intention of listening to see if that fixed the fan noise. Well, now it simply will not turn on at all.

Currently, as soon as I plug the wall outlet cable into the power supply, the computer begins to start up, the green light on the PSU lights up, the white light on the power button lights up, and all seems to be going fine. However, the display never starts up at all, and after maybe 15-20 seconds, the computer turns back off. It then stays off for 5-10 seconds, then tries to start up again. This just keeps looping and looping until I unplug the power cable.

I did some Googling and tried a few basic things. First, I plugged back in those two headers from the motherboard, hoping that would get it back to the way it was, but no, it still loops. I unplugged the PSU from the motherboard, plug it in, then press the built in self test button on the PSU, and as long as I am holding down the button, the green light comes on and the PSU fan begins to spin. I put the PSU back in the computer, plugged in the 2 plugs into the motherboard. Nope, still loops booting up then turning off over and over. I also tried replacing the 2032 coin style battery on the motherboard, and that also did not change anything. Neither did removing one of the RAM DIMMs, and switching out one DIMM for the other (ie only using slot 1, but trying both of the RAM sticks that I have in that slot.)

The confusing part is that I don't see any indicator LEDs on the motherboard, there are no beeps, the green light is illuminated on the PSU, and the white light is illuminated on the power button. These are all signs that everything is going just fine. But no display comes out, and the computer just keeps turning off part way through booting up and then trying to boot up again, over and over.

This computer was turning on just fine a couple days ago. It was on for a couple days, I turned it off a day or two ago, and the only things I had done since then was try to unplug those two headers on the Motherboard.

Here are a couple pictures inside the computer. Any ideas of what else to try?