Well one reason is that past restrictions didn't really work (see Assault Weapons Ban). Another is that current methods of background checks and such don't seem to do anything to prevent mass shootings either since basically all of the shooters lately passed the background checks. IMHO we already have too much of a security state going on with NSA evesdropping, etc. and the amount of additional surveillance needed to "stop" mass shootings would turn us into an outright police state. And it wouldn't be limited to just gun owners either, EVERYONE would be caught up in it. Notice how everyone's info is routinely harvested by government in the name of "having dots to connect" when 99.9999999% of us will never have the slightest thing to do with terrorism.
I suppose if you like security theater like the TSA making you throw out your small bottle of hand sanitizer "just in case it's a bomb" that expanding background checks further are useful as a panacea. Of course those supporting the typical set of gun control measures don't bother addressing the actual concerns of gun owners (like cost, speed, convenience) if "universal background checks" were adopted and miscast it as opposing all background checks generally.