Whether they were right of wrong holding that his specific weapon did not serve a military purpose, they held that his protection under the 2nd amendment was not dependent on his participation in a recognized militia (an idea that the state had presented, I believe) and that his gun was only not protected due to it not serving a valid military purpose.
My contention is that an AR-15 whatever else it may be good for is arguably the single most appropriate rifle for use by a militia and is therefore explicitly protected according to US v Miller.
I’m not sure what his shotgun looked like but I always pictured it as a double barrel sawed nice and short rather than as a pump of a logical but illegal length like the military was using at that time.
You're definitely quite close in your summary of Miller.
The government did claim that membership in a Militia was required and by the nature of their decision the Court did not concur. If membership in a Militia was required to be protected by the Second Amendment, then there would have been no reason to consider the particular weapon possessed by Miller.
The Miller Court, however, did not decide the issue of the usefulness of a short-barreled shotgun. They mentioned the lack of "judicial notice" regarding the gun, meaning that the matter hadn't been addressed; mainly because the lower court considered it an "arm" and simply ruled that Miller himself was protected by the Second Amendment.
The fact is that the Miller Court REMANDED the case back to the lower court with only the nature of the weapon as an issue.
The lower court never again took up the issue, I think, due to the death of Miller and the disappearance of his co-defendant (or was it the reverse?).
For 70 years the lower courts blatantly LIED about the Miller decision and allowed the nonsensical "collective right" notion to take hold.