Wow. I fixed the problem.
It was actually exhilarating, and I feel like I've accomplished something epic. Internet was not much help, and this problem wasn't widespread so I was pretty much alone.
I feel like screaming out 'hurrah' or something a few times. I now have full OGL capabilities after months!
What I did was use Process Monitor to run a variety of programs that used OGL, and where it would fail to find a file. After all, that was the problem. It couldn't find a module (dll) of some kind.
So after running a few OGL programs with the same "Loadlibrary failed" error every single time, I noticed that it would load dozens of dll files. I had to go through hundreds of lines of logs, as even after filtering, it was hard to miss.
I kept searching, hoping to find a .dll file that the OGL program would fail to find completely.
Then I found it! Even if the program couldn't find a .dll file initially, it looked in other folders and found what it was looking for. But this specific dll file would not be found. Any OGL program would try to find this dll file and it would fail to find, thus my horrible LoadLibrary failed error.
The file was simply ".DLL". Any OGL program would look for this in a variety of folders and fail, since it was not there. This is very strange, as it's a dll file without a name. I decided to use the command line and copy and rename opengl32.dll into ".dll". My OGL programs no longer crashed, but it was the wrong DLL or something, the program would error out again. So, I've isolated the problem! After remembering the old suggestions about missing "atioglxx.dll" file, I copied and renamed THAT file into ".DLL" into /windows/system.
I then ran my OGL programs.... And they magically worked! I think I've fixed the problem. Only took a few hours of my day.
Here's the important technical part:
Now, why would after installing 10.x drivers, would OpenGL programs start looking for ".DLL"? Copying over and renaming atioglxx.dll as ".DLL" seems to fix it.
What's the best way to tell the Catalyst Crew about this problem?