Endor, FWIW I bought an Uzi machine gun when Bill Clinton was president. I bought it over the internet and paid for it with a credit card...then waited 8 months for the government paperwork to clear for steps from the manufacturer to the FFL to me. But it WAS an internet purchase which happened in the midst of a Democratic regime and in the midst of the AWB!
An Uzi machine gun is as much the same as an Uzi carbine as a NASCAR race car is the same as your grandmother's Ford Taurus. It looks the same on the outside, but the inside is different, different sizes, different shapes, different parts. Sure you could make a race car from your granny's car but it would end up that all you'd have the same when you finished would be the outline of the vehicle in 3-d space.
Concerning your argument, I once had a similar argument with my barber, a guy I'll call Joe. I told him I'd give him a thousand dollars if he brought me one single news report of a citizen using a legal Uzi to commit a crime. He tried but never could, because the only one occurred a long time ago, 1983 if I recall correctly.
Uzis were popularized as being eeevvviiiilllll rifles in a very weird way. Either Emilio Estives or Charlie Sheen, I forget which one, appeared on the Johnny Carson show back in the late 70's or early 80's. (I forget the specific details, being an old f*rt, but I saw the show.) The kid spent about fifteen minutes talking about a drunken fight he'd had with his brother which involved death threats and either waving or shooting an Uzi at the other. Of course the press ignored the propaganda potential of spoiled children of Hollywood stars shooting at one another but they latched onto the idea that the Uzi was some sort of instrument of death from the bowels of Hell.
In a few years when the dust finally clears and people start counting their change there is a pretty good chance that President Obama may become known as The Great Absquatulator
. You heard it first here on TFL.