The fact is that is a false dichotomy.
It is indeed a false dichotomy, which with respect, I believe you have exaggerated above
Nobody on either side maintains that "Assault Weapons" do not exist.
The U.S. Armed Forces define assault weapons as, "...short, compact, selective fire weapons that fire a cartridge intermediate in power between submachinegun and rifle cartridges." (Small Arms Identification and Operation Guide p. 67)
Modern examples would include the M1A4, AK74M, AK12, HK MG36, SCAR, SAW, RK62 etc., etc., etc.
Not only has civilian ownership of these weapons has been strictly regulated since 1934, the importation of some of these examples is completely illegal.
The dispute arises because advocates of gun control have appropriated term "assault weapon" as a definition of convenience to suit their agenda.
It is patently ridiculous to claim that a rifle which the military does not consider to be an assault weapon is in fact intended for warfare, but as with all things, the people who know the least about a subject consider themselves to be the most knowledgeable