Pragmatically, this was neither a mental health nor a criminal justice issue. With puberty comes some level of social readjustment, and there is no criteria which can tell you what any are in need of in order to adjust. Are you going to put goths in jail because they dress in black? Are you going to put nerds in jail because they might be the next Adam Lanza? Are you going to lock the "promiscuous" girls in an institution until they mend their wicked ways?
Rodgers was a high functioning autistic whose basic juvenile routines no longer worked and so he needed to shift paradigms. This is not necessarily clean, and extreme behavior in order to unearth failure points which can break down the previous paradigm as well as form the foundation of the new, more robust paradigm is expected. (If your theory of physics isn't working you may have to smash a few atoms to find out why.) The problem is there's no way to tell angst from someone whose first go assembled the /pol/ hivemind as a self-reinforcing entity. Rodgers latched onto a worldview that was completely defined in terms of himself, but you're going to find that most people define a heck of a lot of their worlds in terms of themselves at that age. There'd be no way to tell that his first reformation was so pathological that the place where the rubber met the road for the next test wasn't the mere flirtation with the idea of murder or the "crazy" persona, but actual murder.
Neither a hospital nor jail setting would have helped him overcome an autistic social dysfunction. Both are systems that operate by easily identifiable rules and regulations -- each person has their place, and everything is done for a specific function. He would have taken to the routine and looked perfectly healthy (other than normal autistic tendencies), but upon coming out he would be in the same situation as when he went in, only more alienated than ever. A system of institutionalization based on nothing but "worry" would probably create more killers than anything.
Some change to the gun culture would probably help. Americans like their entertainment simple, with morality spoon-fed to them as a black and white proposition with the correct answer always being, "violence." A gun's simplicity of use lowers the bar as to what constitutes a worthwhile killing. (Go through your movie library and in each instance of gun violence ask yourself, "Would the perpetrator still kill if he had to pounce on the victim, overcome them in violent struggle, and chew through their jugular to do it; or would all that effort make an alternative solution the [now] obviously preferable one?")
Steeped in a culture where the self-insert hero is always right and the forces against him are designed to be clearly wrong, it's not surprising that conservatards are as racist and self-absorbed as they are or that someone like Rodgers likewise might get caught up in such a tale.