Ok so I did both, but the second one got me a bit further/showed that something more fundamental might be wrong.
What I mean by that is that after I deleted the masseffect.exe in the binaries folder (while origin was closed) and then re-opened Origin, Origin hadn't even recognized that it had been deleted, it just tried to install ME1 like normal and once again said complete even though I very much knew there was a missing file.
So now I am supremely confused as to why Origin didn't even recognize that it had been deleted.
I'm going to independently try to delete it and download it again now that I have safe mode on if that'd make a difference, but I am stumped.
Thanks for the help and if you think of anything else I'd really appreciate it.