I had today off from work so I spent a little bit of time wrenching on the Ghia, and I think I figured out why the starter doesn't start when I turn the key to the start position - and probably why the previous owner had hooked up the silly push-button starter setup in the first place.
It seems that the 1974 Ghia includes a seatbelt safety interlock which prevents the car from being started unless everyone is securely buckled into their seat. Since I already had the seats out of the car, it was easy to hook a multimeter up to the connector under each seat and determine that the passenger side seat was registering an occupant (closing the circuit) even when I wasn't sitting on it. I then checked the wires coming off of each seatbelt buckle and found that the passenger side was (of course) thinking that the seatbelt wasn't fastened even when it was.
So the car always thinks someone is in the passenger seat, and never thinks the person there is wearing their seatbelt.
It seems like a pretty complicated system; wiring up the push-button starter was probably significantly easier than correcting the faults with the seat and buckle. I'd still like to have things working the right way but I can at least accept the push-button setup (for now) since I better understand the reasoning for it.
With that mystery solved, I reinstalled the seats in the car and went for a quick spin around the block. It was only about a 5-mile drive, but I drove past three elderly men who each gave me a wave or thumbs-up gesture as I went by. That's pretty cool.
That's the extent of the car work I did today; I also just picked up How to Keep Your Volkswagen Alive by John Muir and have started reading through it. I can already tell that this will be an excellent guide to understanding this new-to-me air-cooled and carbureted engine setup as well as offering practical instructions for keeping it running. I think it will compliment the more technical Bentley manual quite nicely. Thanks again for the recommendation, @timapple!
I do intend to take a closer look at that weird steering issue later this weekend; I just really didn't feel like getting underneath the car while I'm home alone. More soon!