Greetings all! Update here!
I fixed it!
Here's what happened:
I replaced the thermostat and filled the system back up with coolant/distilled water. Turned the car on and it still overheated.
Went back to the boards and read some more about how to "bleed" the cooling system. It turns out that if the heater isn't blowing hot air after you turn the car on, that there is air trapped in the system, and that I didn't bleed the cooling system properly.
I decided to go against taking the coolant plug out from the passengers side exhaust manifold area. That left one bleeder screw, 2 inches to the right of the main coolant fill hole in the top of the radiator reservoir.
I looked at the coolent level, which was full to the brim, and then loosened that screw. Within seconds, the level dropped down with a noticeable air purge coming out of the bleeder screw hole.
I then poured a good gallon or so of distilled water and coolant (50/50) into the resevoir until it filled it.
Turned on the car and squeezed the upper and radiator hoses and few times to push out air. I then replaced the bleeder screw and put the radiator cap on.
This time, the upper radiator hose got super hot which meant that the water was indeed travelling through the block and into that hose... unlike before. This is a good thing.
Within about 10 minutes of idle, my fan came on as noticeable heat was now coming off of the engine.
Took the car for a test drive and had no further issues with the cooling system. After letting it sit overnight from it's run, I refilled the resevoir another 1/4 gallon or so (from air moving out of the system) and capped it back up.
No further issues whatsoever.
Turns out the 10 step instructions for bleeding your cooling system is actually important. It's not like filling up an old truck radiator...
