What's a policy that doesn't punish the law-abiding who will never do anything harmful (roughly 13.499999 trillion chances to have happened just this CY alone) and keep them out of the hands of people who a. don't follow laws and b. have sociopathic tendencies?
Positioning the question this way is, in my opinion, problematic. It indirectly suggests no room for compromise for fear of being inconvenienced. In my opinion, there is no solution to gun violence in this country that does not in some way inconvenience the practice of gun ownership. Phrasing the question this way also assumes people will always do the right thing, which we know they don't.
So, I think it's important to look at it from a different angle i.e. what's a policy that saves lives, protects our children, and still allows for law-abiding citizens to own firearms in a responsible format. But if inconveniencing gun owners is more important than preventing children from being murdered in school (which is largely the message being sent by certain elements of our society) then there is no satisfactory policy out there.
Responsibility is really one of the key issues here and we just don't have enough of it. But, I personally think there are a number of solutions other countries have implemented that we could consider implementing or even tweaking to fit our needs. Things like:
1) A more robust application process
2) Requiring a medical doctor/personal physician to approve someone for gun ownership
3) Requiring independent referees (non-family and citizens in good-standing) to act as references on a gun purchase application, to provide statements about the applicant's character, mental state, etc.
4) Proof that the weapon has a secure location to be kept in the home
Another option (which would never happen here, but I'll mention it anyways) is requiring someone to provide 'good reason' for owning a firearm i.e. profession, sport, hunting, etc. For example, self-defense or 'I just want one' wouldn't be a good enough reason. Again, I wouldn't get too held up on this particular point. I don't think it would ever happen here. But, I do think some variations of the aforementioned options are worth consideration as they have been found quite effective as part of a comprehensive solution in other countries.