The op thinks it's different because a part with a serial number is installed into a non nfa weapon, and it becomes one, a machine-gun.
Why can't a part be installed onto a pistol to make it an sbr?

Having just seen this, at 11:30 pm, I have not made any inquiry or read the section in the nfa handbook, but I think I can pretty well answer this from what I already know.

Machine-guns are defined by one attribute only, the discharging of more than one shot by a single pull of the trigger. Nothing else matters.
If only one part makes it work as a machine-gun, then that part can be the machine-gun, and be registered and numbered.

However, short barrel rifles are determined by several factors. One, it must be a "rifle", ie., designed or redesigned to be fired from the shoulder. Therefore, it has a stock. Two, it must have a barrel measuring less than 16" in length, an overall length of less than 26", or both.
So, it must have both a stock, and a barrel or overall length under the, minimum requirements to qualify. While you may add only one part to finish the qualification, several parts are involved in the qualification.
Plus, a pistol cannot readily accept a stock, so in order to add a stock to an AR pistol, for example, several parts are needed. The buffer tube alone would instantly qualify it, because now it readily accepts a stock. But, there still is not a stock actually on it.
Basically, gunsmithing is involved, whereas the drop in auto sear or bolt is merely a part change, not breakdown of parts requiring tools, as in the removal of a buffer tube.

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