This seems to have worked...sort of.
Performing a clean boot did allow me to install and run Origin. When I boot normally, however, I cannot run Origin even though it is installed, meaning I still can't run the game. Performing another clean boot once again allows me to access Origin and, in turn, to play the game.
I guess this just means I'll have to perform a clean boot every time I want to play Mass Effect. Definitely a lot better than not being able to play it at all, so thank you kindly for that.
If you have any tips that might allow me to run it without the clean boot, I'd appreciate it, but otherwise my problem seems to be solved.
EDIT: I decided to try re-enabling things bit by bit until it stopped working again and what I found was...unexpected. I was able to re-enable all the msconfig services (all the ones I recognized, anyway) so then it was time to start messing with the Task Manager startup items. As it turns out, my Huion Tablet utility, for WHATEVER reason, was the thing not playing nice with Origin. So...a warning to anyone who uses a Huion tablet and wants to use Origin, I guess.