It was the NRA and activist members, who got handguns removed from the act.
Licensing and registering all handguns was the major thrust of the NFA bill. Machine guns, silencers, and short barreled rifles and shotguns were just the consolation prizes left after the bill was gutted.
The NFA was proposed during the quaint time in history when Congress was under the impression that they couldn’t Constitutionally actually ban anything, but could effectively ban things by taxing them heavily.
I don’t recall the Constitutional Amendment that lets Congress pass laws on anything/everything but it must be a big one.