I am by no means a fan of Trump's but he's no moron. He has a hugely inflated opinion of himself which is what both gives him the confidence and bravado to confront what he perceives to be evil or wrongdoing fearlessly but it's also (IMO) what drives his need to exaggerate and make himself out to be #1 in everything, to inflate his own deeds even when they are substantial even without exaggerating.
I fully believe Trump probably would have gone in after the shooter because he believes he is invincible and nearly invulnerable. He's gotten where he has in part because everyone else keeps mistaking his braggadocio for stupidity and assumes he's just being vacuous and over-the-top when he says something.