While a criminal background check may prevent and remove some firearms, it won't stop things like San Bernardino. In order to stop that latest shooting, we'd need a Minority Report-style pre-crime analysis of people, which not only isn't technically possible, it's unconstitutional.
Back to your original point: California already has a system that automatically flags gun owners who commit prohibiting crimes and disarms them, the California DoJ Armed and Prohibited Persons System, aka APPS. They go after 5150s as well. However, they often overstep their bounds disarming spouses as well, until lawyers reign them back in (the firearms just need to be secured from the Prohibited Person, which can mean they just don't know the combo to the safe).
Additionally, CCW holders in California are constantly being watched by their Issuing Agencies. No doubt their IA will know very soon after an arrest and revoke their permit and secure their firearms. At a minimum, California CCW holders go through a background check every 2 years during the renewal process.