Something needs to trigger the mental health exam. Do we depend on the shooter himself or his family? The track record on that is extremely poor. Outsiders? What gives them the authority? Who provides these exams? Who pays for it? At what level is it OK to take someone's rights away? It's a solution that sounds great in theory but had too many questions that need to be answered if it is to be implemented.
No plan is perfect. Will some slip through the cracks? Yes. Will enforcing the laws on the books help? Absolutely.