I guess if we want to restrict to that–a number probably 4-5x higher than we have now, maybe even more so–at the very least mass shootings would need to represent responsibility for a majority of homicide deaths in America, and even then, it will likely have to be much, much higher. The reason being at core, we simply don’t associate those illegal gun uses with our legal gun use, but there is probably some very high level of death that would carve off enough casual gun owners and convince them to try gun restrictionism (which I think would likely be meaningless at addressing the problem.)
The premise of the question in thread title is essentially flawed because it makes an association between events and gun policies that gun owners simply do not accept–you can disagree with them not accepting it, but that doesn’t change their minds.
It is my believe that if you want to work with gun owners to try and pass some gun regulations, you have to find a way to associate with gun owners on values that gun owners hold. The premise of “we need to add restrictions to you owning a gun, or prohibit certain types of guns, because bad people do bad things”, is never going to persuade many of us. However, a premise like this, “responsible gun owners know how to safely operate a firearm, and that is an important responsibility of gun ownership, so we think there should be a relatively straight forward safety class you have to take before buying your first gun.” That is going to get a lot more traction with gun owners, appeal to their responsibility to be good gun owners, instead of attacking the concept of gun ownership, and you’ll at least make some level of progress.