For a moment there I was thinking this might have all been for naught. Turns out though, I just didn't have the 20 pin connector snugged in tight enough. It does run! How embarrassing, but by admitting these small snafus we can all learn to not feel stupid by simple mistakes.
There was also a brief moment where I thought the power supply wouldn't fit, and I was genuinely confused as to why, but then I remembered that I had bent the tab that the psu normally sits on backwards to try and fit the old Dell psu. There's a picture below where you can see the rectangle-shaped cutout toward the back of the case in the upper-righthand corner. That's where the tab bends forward into the case to act as a little shelf for the psu. It's really not needed, but it does help you guide the psu into place so that the mounting screws are easier to line up.
So I was sort of right in my guesstimation about which, or what, parts I had. Unfortunately, about the only thing you can swap out of a Dell are the hard drives, PCI cards, and maybe the memory sticks. I thought I could get away with cannibalizing the power supply, which would have meant that I could've gotten this Green Frankenstein up and running last night... That didn't happen.
I've desperately wanted a reason to build a Slackware server for quite some time now. Just recently, someone on Mastodon finally gave me a great idea for what I could do if I ran one. There is a bunch of old hardware sitting in my garage just collecting dust. As much as I would love to buy some refurbished server hardware to play with, at the end of the day I'm not going to be doing anything too special with whatever hardware I use. So I may as well try to just re-purpose the stuff I already have.