You're welcome!
So if I understand correctly, since you have Origin EA play, and can thus dl the DLC on Origin, I don't see why you wouldn't be all set. Once you have the DLC in the correct folder, you should be able to play MP, whether through Origin or EA desktop. I can.
You said "all the DLC's are gone".
Do you mean gone from your PC folder structure where you previously installed the DLC? Or just "gone from the store"? If it isn't in the structure, that is odd. Did you install ME3 to a different drive than before?
If you mean just "gone from the store", shouldn't matter. Just install from Origin.
And FWIW, I don't see EA desktop as any less "janky" than Origin. Nor do I have high hopes for it being so. But I don't have major issues with Origin. I think I've had more with EA desktop. So yeah, just play from Origin.