From Richard on Patreon:
"Why didn't the M3 and M3A1 (grease gun) have a double stack/double feed mag and use the Thompson mag? It would have certainly simplified logistics."

The answer to the first part is that the M3 used a single-feed magazine because it was largely copied from the Sten. A new magazine was necessary, and the Sten design looked like a good enough design that was inexpensive and worked well in a stamped SMG.

The answer to the second part is that the Thompson magazine is retained in a gun in a way that is not compatible with a simple stamped design like the Sten or M3. The Thompson has a large T-shaped lug on the back of the magazine that slides up into a matching track behind the magazine well. The Thompson has a big milled block of a receiver that can accommodate this track easily. Several other US SMGs did actually copy or use the Thompson magazine (the M2 and the UD-42, specifically), but these were also designs with blocky solid receivers. The stamped design of the M3 (copied from the Sten) had no such block of material where that sort of magazine retention could be placed.