I won't get into all of the effort to remove the water pump pulley, crankshaft pulley, etc, but I did get the radiator out and everything torn down enough that I was able to remove the timing belt. I spent a good amount of time getting all of the timing marks lined up, then removed the old belt and installed the new belt. Everything looked fine, so I moved on to the water pump. Installed a new alternator belt as well, and then installed the new radiator.
Feeling very pleased with myself, I filled the radiator up and eagerly jumped into the drivers seat to start the car so I could start the car and finish topping up the radiator.
Car would crank, but would not turn over at all. I tried until basically the battery was drained, so I threw the battery onto a charger and gave up for the night.
Next morning, I put the battery back in, crossed my fingers, and tried again. Same issue. Car cranks, but it doesn't sound like it's trying to spark at all.
So, back to reading online to see if I could diagnose the problem. I thought perhaps I had inadvertently damaged my coil, so I tried grounding the coil wire to the engine and confirmed spark. I then removed a plug and verified spark there as well. I then moved over to the carburetor and verified the fuel pump was working and fuel seemed to be making its way into the cylinders (as evidenced by the wet plugs I had to clean up after cranking the engine for so long).
I decided to re-verify my timing, so I moved everything to TDC, and noticed that my crankshaft pulley was out of alignment, so I loosened the belt and re-adjusted so everything was properly aligned. I then removed the distributor cap in order to verify it was pointing to the #4 plug. This is what I found:

It appeared that the distributor was out of alignment, so I adjusted that as well. Buttoned everything back up and tried yet again to start the car. No luck, and I am at my wits end at this point.
Does anyone have any other suggestions for me before I tow the car to a mechanic?