I believe there is far more to it than that.
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.
The key here is "shall not be infringed." If you look up the definition of infringed:
1) to encroach upon in a way that violates law or the rights of another
The way I see it, anything the government would do that would impede citizens from owning firearms is an infringement and therefore un-Constitutional.
Merely saying they cannot take away the right to own a firearm doesn't go far enough to protect the right. By that definition the government could make it extremely difficult, costly, and time consuming to own a firearm - all while claiming they are not absolutely preventing ownership.
Anything that adds cost (taxes, background checks) is an infringement because it raises the financial bar on ownership.
Which is the express purpose of the NFA of '34 and every gun "regulation", tax, and arbitrary edict to come out of the FedGov and the States.
The BATFE should be disbanded with prejudice. Immediately.