Mac uninstall, now game will not open at all and files are gone from The Sims 3 file

The folder you are describing - a folder called "The Sims 3" within a folder called "Electronic Arts" is within Documents section of your Mac, right? This is where the game stores files when you are playing (such as saves). These files are created when you actually play the game. You could in fact delete that folder and then when you start the game it would make a new one. So the fact that it is empty may not be the problem. However if you look in the Applications on your Mac instead you should see another folder called "The Sims 3" ( this one is on it's own, not inside another folder). This one in Applications contains the actual game installation. If you look in there you should see shortcut icons for the base game and any expansions you have installed. Does it look as you would expect? If not then your installation from Origin has failed.

Can you actually launch the game at all?