You actually can launch and play it from the EA Desktop once you have the DLC's installed in the proper location.
But, yes, you can't get the MP DLC's from EA Desktop app, so if that's your only option (i.e. you are using Game Pass), and can't find the DLC files elsewhere, you're out of luck.
"once you have the DLC's installed in the proper location"
And where is that proper location?
Under your Mass Effect 3 install folder, it will be in:
\BIOGame\DLC
Each DLC has its own sub-folder starting with "DLC_"
The default location for your game installs is C:\Program Files (x86)\Origin Games\
In case you changed, and need a reminder, you can check the properties on the game's startup icon. Or from inside origin, from the menu at upper left of screen, check:
[menu] Origin>Application Settings>Installs & Saves
Yeah, that is exactly where they are and I still can't launch ME3 from the EA app. It gets stuck in an endless "update required" cycle
