Okay guys! I finally solved my own problem, thanks to some minor research that keyed me into believing the circuitry on the home button flex cable was the problem. I reopened my iPad (after a considerable amount of time mentally preparing for the agonizing task), installed my original Apple home button from my original broken digitizer, and it completely fixed the problem!
Apparently the tiny amount of circuitry on that flex cable is responsible for recognizing when the Smart Cover is closed and open.
Thus, if you want your iPad to function correctly with a Smart Cover, and you don't want it to lock your iPad when it's laying flush against the back of the iPad, I recommend always transferring the original Apple home button to the new digitizer.
The cheap OEM parts that you can buy on eBay are always fraught with problems. It's possible that 90% of the time an OEM digitizer from eBay will come with a perfectly normal home button. But Simon and I probably both ended up with bad ones. If you can salvage the home button (and its flex cable) from your original Apple digitizer, do it! :)