I think you need both visible security and teachers carrying concealed. It should be optional/at will for teachers, but be required of Administrators.
I also feel there should be mandatory training firearms, shoot, don't shoot etc for those who do carry. There should also be mandatory training for all school personnel on what to do in case of an active shooter, bullet proof secure doors on all entry and classroom doors and mandatory drills carried out with students.
That way panic would be less, in a lock down you could secure your door so no one could get to you and as you say, worst case scenario an armed response would be available.
This would go hand in hand with counselors and psychologists being required to forward the names of all students who they know pose a potential threat to others to the authorities. The school system should then required to have a hearing on whether to remove the student or not, as well as a judge review whether the threat should be put in mental health custody, barring a complete psychological review. All of the above should be statutorily required with a very short time window allowed.
Finally, counselors, psychologists and administrators should be criminally and civilly liable for any failure to comply.
There's a long answer to your question.
P.S. Bullying should be an automatic expulsion and the teachers/admins civilly liable for failure to deal with it.