You are correct.
Every school shooting is a tragedy. It is a terrible, awful, awful thing. As a parent, I can't even imagine...don't want to. But it's really, really rare. Since 2000 there have been an average of 2 incidents per year of gunfire with injury at a K-12 school. Most of these only involve one victim, sometimes the shooter himself. There are 100,000 public schools in the US (and another 30,000 private schools). There's an average of 180 school days to a year. So at US public schools, 18,999,998 times per year a school bell rings in the morning, and in the afternoon, and in between not a shot is heard. And 2 awful, awful times, there is. But it's a 1 in 9 million chance of it happening on any given day. And even that tragedy injures or kills a very, very small fraction of the students.
There's just nothing you can do to prevent an event that rare that will not have unintended consequences, and those unintended consequences may be worse than what you were trying to prevent.
Ban all guns? First, good luck. There are something like 300 million guns in the US. Who's going to collect them all? And there's the unintended consequence, that the criminals who don't turn in their guns will now know their law-abiding victims are unarmed. More people are going to die because they can't defend themselves. And we already have background checks and laws against selling guns to criminals, the mentally ill and minors. The shooters get their guns from people who wouldn't be on a list denying them purchase anyway.
Arm teachers? First, teachers are not cops or soldiers. The Kindergarten teachers are not going to suddenly morph into SEAL Team 6 and take out the bad guy. More likely, a student will get his hands on a teacher's improperly secured gun, or the teacher him/herself might be the one to go nuts and start shooting. Now you've got a shooting where there never would have been one otherwise.
You can spend millions turning every school into a fortress, but it's still not going to stop a determined attacker. It's the defender's dilemma. Plus the opportunity cost...you could have actually spent that money on teachers and books instead.
There is this fetish for control in America, where people seem to think you can make it so nothing bad will ever happen. But you can't. The world is messy. There's nothing you can do to stop events that rare. Mourn the dead, hug your kids, watch out for troubled youngsters who need help, and move on.