There is a "real reason". What you deride as "just" an "NRA slogan" is actually a core finding made by Justice Scalia and the majority in
Heller:
"We may as well consider at this point (for we will have to consider eventually)
what types of weapons
Miller permits. Read in isolation,
Miller’s phrase “part of ordinary military equipment” could mean that only those weapons useful in warfare are protected. That would be a startling reading of the opinion, since it would mean that the National Firearms Act’s restrictions on machineguns (not challenged in
Miller) might be unconstitutional, machineguns being useful in warfare in 1939. We think that
Miller’s “ordinary military equipment” language must be read in tandem with what comes after: “[O]rdinarily when called for [militia] service [able-bodied] men were expected to appear bearing arms supplied by themselves and
of the kind in common use at the time.” 307 U. S., at 179.
The traditional militia was formed from a pool of men bringing arms “in common use at the time” for lawful purposes like self-defense. “In the colonial and revolutionary war era, [small-arms] weapons used by militiamen and weapons used in defense of person and home were one and the same.”
State v.
Kessler, 289 Ore. 359, 368, 614 P. 2d 94, 98 (1980) (citing G. Neumann, Swords and Blades of the American Revolution 6–15, 252–254 (1973)). Indeed, that is precisely the way in which the
Second Amendment ’s operative clause furthers the purpose announced in its preface.
We therefore read Miller to say only that the Second Amendment does not protect those weapons not typically possessed by law-abiding citizens for lawful purposes, such as short-barreled shotguns. That accords with the historical understanding of the scope of the right, see Part III, infra.
25" (Emphasis added)
https://www.law.cornell.edu/supct/html/07-290.ZO.html