I think they would be used about 1/20th as much as rifles, which are used about 1/20th as much as handguns. And I think you have the same reasoning behind the reductions. Handguns are easier to use than any rifle. easier to conceal and carry as well. Cheaper, both in the upfront cost and the cost to shoot. automatics have a similar jump in complexity and price over rifles as rifles do over pistols.
so yes we would see them, but I don't think the deaths would necessarily go up as much as you think. Its not like the movies, most people aren't holding a fully auto weapon on target very long at all.
its comes down to the simplistic of economy, automatic weapons are EXPENSIVE. both to buy and to run. rifle ammo isn't far off $1 a round. I think it was a Chris Rock joke about putting bullets on layaway.