It was simply referencing existing case law aka United States v Miller.
The language doesn't need to be exact because that paragraph has no standing.
I agree it should be clearer but it is written and it isn't going to change.
No lawyer is going to use an omission of the word own to indicate it means you can own when Miller clearly indicates you can't own NFA items without them being registered and the NFA doesn't violate the 2nd ammendment.
Bad for us is the fact that Miller had a sawed off shotgun.
Per Miller:
The National Firearms Act, as applied to one indicted for transporting in interstate commerce a 12-gauge shotgun with a barrel less than 18 inches long, without having registered it and without having in his possession a stamp-affixed written order for it, as required by the Act, held:
1. Not unconstitutional as an invasion of the reserved powers of the States. Citing Sonzinsky v. United States, 300 U.S. 506, and Narcotic Act cases. P. 177.
2. Not violative of the Second Amendment of the Federal Constitution. P. 178.
The Court can not take judicial notice that a shotgun having a barrel less than 18 inches long has today any reasonable relation to the preservation or efficiency of a well regulated militia; and therefore can not say that the Second Amendment guarantees to the citizen the right to keep and bear such a weapon.
I always wonder what would have happened if Miller had instead been caught with say an assault rifle. A weapon that clearly has a purpose in a well regulated militia. It is possible the courts would have scrapped the NFA.
An omission in Heller of something explained in Miller doesn't equal a reversal of the courts.
Now if it specifically said you can own but not carry that would reverse Miller but that isn't what happened here (it would also have been outside the score of the case).
Likewise if Miller didn't exist the statement just carry (no mention of own) would likely be grounds for a future suit in which Heller is referenced as saying "carry" but not "own".
Since regulations on NFA items were found Constitutional in Miller that stands until directly overturned in a future case.