I wanted to make a new thread so people could easily spot it.
The fix is simple: Reinstall Origin.
The reason this bug occurs is because there is a problem detecting Origin. If you go into the Program and Features tool, you'll notice that Origin is not listed anywhere. The Sims 4 demo thinks Origin is not installed and therefore refuses to start the game. Reinstall Origin and make sure it is listed under Programs and Features in your Control Panel. I'd reinstall the demo just to be safe as well.
How to fix the problem:
1: Uninstall the demo. Right click the picture in Origin and hit the uninstall button.
2: Go to Control Panel. Then hit Programs. Then hit Programs and Features. Search for Origin. It shouldn't be there.
3: Uninstall Origin with the executable. The default location is here: Local Disk (C/Program Files (x86)/Origin/OriginUninstall.exe
4: Reinstall Origin by clicking here
5: Go to Control Panel. Then hit Programs. Then hit Programs and Features. Search for Origin. It should now be there as the most recently installed program.
4: Reinstall the demo
Steps 2 and 5 are important in verifying that this is indeed the problem causing the demo to fail, that The Sims 4 is bugged into thinking Origin isn't installed when it really is.
I hope this information helped EA and Maxis as well to prevent this bug on launch.