Actually, you basically solved this problem at the very beginning. The only thing keeping you from putting it completely to rest is my own stupidity.
When I unchecked every background task running in MSCONFIG, the crashes stopped. But when I turned everything back on, the crashes did not return which I thought was odd as I figured they would. For several days now I did not see the BEX64 crash. I restarted GC many times while messing with Game Link and not one single crash occured.
Early this morning, I decided to load one of my games. I immediately noticed my framerate was not being displayed as it should be. I looked in my system tray, and the program to display my framerate was not running. I then realized that I forgot to turn on this one program when I turned everything else back on while trying to figure out the BEX64 error in GC. Right then I realized that maybe this program is one of those overlays you mentioned that can be related to the BEX64 error.
With this program running again, I turned on GC 4. It crashed almost immediately. I turned the framerate program off and ran GC 4 many many times with no crashes. I turned the framerate program back on and GC crashed almost immediately. Turned frame rate program off again, and GC worked fine.
The program causing this issue is FRAPS. I googled the BEX64 error (something I should have done at the beginning but never did for some reason) and sure enough, overlays are typically what causes this particular error as you already discovered. On more then one webpage, one program that is listed as a potential cause of this crash with games is Fraps. Fraps does think GC is a game as my frame rate is displayed in GC when fraps is running.
You mentioned overlays to me in that other thread right at the very beginning but I simply did not make the connection between overlays and Fraps. The ONE program I forgot to turn back on happened to be the one program causing the issue. Go figure.
I already tried fraps + GC 4 on another computer and there are no issues on that machine so it is very likely just happening to me on this one computer. I do not need fraps and GC 4 running at the same time, so I can simply turn off fraps when using GC.
Thanks for reporting problems here.