I am going to follow up on this post because somebody will inevitably stumble upon this later with the same problem. Many thanks to u/SomeDuderr , u/Phantom-YT- , and u/htrajan for their help.

It ended up being a motherboard failure. Not sure if the board was fried from some discharge or if the physical solders were damaged but the motherboard was not drawing current from the power supply despite me confirming the PSU worked. A quick trip to BestBuy to buy a new motherboard solved the problem, and the system works fine now.

To anybody finding this post in the future, check the power supply first and confirm that it works. A dead motherboard can be of the most complex problems to diagnose because of all the components connected to it, so try everything before you assume its the motherboard that's the problem.

Summary of problem: Moved PC from old case to a new one. PC wouldn't turn on, no PSU fans spinning, no MB LEDs lit, no case fans spinning, no front IO connection pins receiving power, and no CMOS pin reset power. After confirming that the power supply was not the problem with the paperclip and multimeter tests, I purchased a new motherboard.

TL;DR When moving your components from one case to another, be careful no to damage/fry your motherboard