Although this would certainly address some of the possible issues and most definitely curtail the killings somewhat (from those with a criminal past, certainly), I don't think what you call for will cast a wide enough net on individuals such as Eric Harris and Dylan Klebold or Adam Lanza or Christopher Harper-Mercer. Did any of them even have a previous criminal record? Did they ever do anything to warrant the stipulations that you suggest? It seems evident to me, that the only way to stop these individuals is to eliminate their access to guns, period.
But how can this be done when a large enough percentage of the population are willing to accept these atrocities "as part of living in America," for one, or that the lives of innocents are the payment they are more than willing to make in order to keep their guns?
The answers, unfortunately, are quite sobering.
Peace!