AAAYYYY!!! Finally got the gaming rig back online! Spent a solid week trying to figure out what was wrong, searching down codes and guides and unsuccessfully updating drivers. Everything was pointing towards the network card not being connected, which is a serious issue since it's soldered to the motherboard.
But, I opened it up yesterday and gave it a good, thorough cleaning. I also made a couple small changes I've been wanting to make, like switching one of the GPU's VGA cables to another rail on the PSU and flipping the water block. During the initial build, the lines for the watercooler were just about pressing on the upper fans. I didn't realize until after the fact that the block can be rotated and the piece with the logo on it comes off to also rotate, so it's never upside down.
It'd been about a year since I built the rig, so it definitely needed a cleaning. Thankfully, I didn't wait so long that the dust had started to cake into it and it all just blew out without any need for brushes to knock it loose. Flipping the water block also gave me an opportunity to apply some fresh thermal paste, which it needed as the old stuff was looking a bit crusty and dry.
I'm not sure exactly why cleaning it worked, but I'm glad it did. I'm thinking maybe it just needed some time fully unplugged to reset some things (because computers are finicky as shit) or there was a piece of dust that happened to bridge the wrong gap. Either way, it's back up and running and cleaner than ever. And I didn't have to buy a new network card! :D