Most death are caused by handguns. We'd be better off limiting those to people with a concealed carry permit, and concealed carry should require a pretty strong background check. People with conceal carry permits have never been involved in a mass shooting as far as I can tell.
Handguns account for 71 of 143 weapons in mass shootings. Handguns are used in most murders.
Limiting handguns makes sense in other events. People won't carry a rifle around city. Individuals in their own home could defend themselves with shotguns or other rifles. They are actually very popular for that use.
If you limit handguns to those with conceal carry permits, there would be less of a second amendment backlash because gun nuts with rifles won't feel like the government is trying to come and take their defense mechanisms.
Furthermore, one need not limit all types, only automatic ones. Someone isn't going to go on a shooting spree with a revolver, but they could defend themselves with one.
An "assault rifle" ban would only have made 48 of 143 weapons illegal, some of which were hand guns because of the large magazines.