Here - specifically - is where you went wrong. To be "classified" as a pistol, the first portion of your sentence, it cannot have a shoulder stock attached, as in the second part of your sentence.
So as I said above, your interpretation as you described it is wrong. A pistol cannot have a shoulder stock. A pistol can be shouldered and fired, and remain a pistol, which was recently established (re-established) by a ruling by the ATF. For many years, the standing rulings were the opposite, such shouldering a pistol was deemed to create an SBR.
Our opinion on deadliness is irrelevant. The law says a rifled barrel less than 16" plus a shoulder stock is an NFA item, an SBR, not a handgun, so indeed, there IS "reason a pistol can't have a stock attached to fire it shouldered." Doing so is a federal firearms violation.