The Grease gun has a far simpler and cheaper magazine catch than the Thompson.
While they probably could have come up with something that would have worked, Ordnance wanted a newer design magazine that was a little tougher then the Thompson mag and a simpler design.
The Thompson and Beretta 38 series magazines are better IMHO. They are double stack, double feed, and plenty tough enough. The M3 apparently copied the Sten mag, which apparently copied the MP.38/40 mag, that is, a double stack single feed. Such mags were well known weaknesses in these weapon designs and its unusual that a weakness would be copied, first by the Sten, then the M3. Such mags also require a loading tool. Double/doubles don't. You can't load an MP40 mag to capacity without a loading tool, or great effort; the others either.
The inherent problem with double stack, single feed SMG mags is that there is friction created when the double stack necks to a single. Think of what happens on an interstate highway when a triple lane becomes a double in high speed traffic Dirt, corrosion, sand, mud, grease, dust, etc. all contribute to the problem and feeding issues. Such feeding mechanisms surely work, but alot more attention needs to be paid to the lips, keeping them clean, follower fit, and spring tension. I know first hand. Why the M3 was not designed for a Thompson double stack/double feed mag is a mystery, but it was a step backward on an otherwise exceptional weapon.