Actually, that was the main concern behind the OTL Second Amendment. Remember that until 1867 (and in effect until the early 1900's because of the Slaughterhouse travesty), the Bill of Rights only applied against the federal government; none of it had any impact on state laws. If a state wanted to ban guns, free speech, or anything else, it might run afoul of its own Bill of Rights, but not the Federal one.
With that in mind, I see this amendment as being somewhat plausible: it's more clearly making the point that the states have the right to themselves provide for the common defense by having their own militias. (Separately, the Founders would be assuming that people can go hunting for themselves, but that's totally a matter for the states they're in.) The interesting question is what the framers of the Fourteenth Amendment would do when they wanted to explicitly protect freedmen's rights against states who wanted to (among other things) disarm them and leave them defenseless against white racists. IOTL, they did that by extending the Bill of Rights against the states as well. ITTL, without an already-existing broad guarantee of the right to bear arms, we might end up with a Fourteenth Amendment that explicitly protects the right to bear arms for self-defense.
(Which might be harder for a racist Supreme Court to read out of existence, so we might have less lynching and other Klan terrorism...)