I managed to make it work but this is beyond ridiculous.
You have to install Origin (not EA App). And it must be a very specific version of it. Then disable the option to automatically update the app because you want to start at a specific version of Origin. You don't want it to update to the point of prompting you to switch to the EA App.
With that specific version Origin I could finally launch the game. This happens for other games like Dragon Age Inquisition, too.
It's absolutely pathetic. I can't believe EA abandons their products this way. The best solution would be to just release the game on Steam and REMOVE the need of the EA App or any secondary launcher.
I got it to work with the EA app by clicking on the three lines in the top left above Home then clicking Settings>Application>Disable In-Game Overlay.
I was having this issue in both Mass Effect 3 and ME:LE.
