(A lot of ableism in this response, for demonstrative purposes, sorry)

"Crazy" is a term with strong negative connotations that refers to people who experience mental illness. When you're not well, it is very common to worry that you're crazy, or that people will think your crazy, or god forbid that people will find out you're crazy.

It's not clinical, but it's a word that can be very hurtful.

why is it a problem to call out behavior that seems psychotic?

Psychotic is a general clinical term denoting several forms of mental illness. If you are not trained in diagnostics and not working with a person to help them it's generally unhelpful to guess at what illness they might suffer from, and as I said, can be upsetting to others (friends, bystanders, people who are genuinely ill and are constantly afraid that people will find out they're ill).

But couldn't "being a jerk" be the result of some undiagnosed mental condition, such as sociopathy?

It is very possible that behavior you dislike or find abnormal is caused by someone's mental illness. Even if you are qualified to diagnose that, reading a twitlonger, looking at a meme someone made, even reading an IRC log doesn't provide sufficient insight to accurately diagnose them.

It is not helpful, and further stigmatizes mental illness, causing harm to people who suffer. Yes, including antisocial personality disorder. I'll get to that in a bit since I think it's something of a special case.

What are the proper terms to insult people who think or act in dangerous or ridiculous ways? Is "idiot" OK? How about "stupid"?

Idiot, I think, is sufficiently divorced from its clinical meaning that you can probably use it safely, but if someone told me it made them uncomfortable I'd listen, and try to find a better word. As I said, good ways to insult people should address their behavior, not ascribe mental illness to them. A little profanity can make it more fun. "Check out this jackass doubling down when someone called them out on their racist shit" is a lot better than "What's wrong with this psycho?"

What about the people who act in ways that are detached from reality?

"He's completely detached from reality" is a good one.

"Gator" would end up being banned for the same reasons.

Gator (in the context you and I use it) is a neologism. It does not describe mental illness, it describes an asshole who supports gamergate. I've seen gators get outraged at being "dehumanized" by the term, but the term describes someone who has voluntarily supported a hateful movement, and can stop doing so at any time, it does not involve armchair diagnosis and does not stigmatize mental illness.

Is it the words we use that is the problem, or what we are trying to describe?

It's structural. It's about equating mental illness with evil and personal shortcoming. If you see someone doing something you don't like, and you associate that with mental illness? That's structural. That's the system (and I've no idea which fucking system, or if it has a name, because it's not really the patriarchy, white supremacy, or classism, though of course all three support it) being expressed through your thoughts and actions. It's reinforcing a system where it's okay to hurt some people because they're different than you. And I don't mean gators, I mean friends, family, and bystanders, because some people legit wake up thinking "I'd rather kill myself than let them know I'm this way." Please, don't hurt those people.

Circling back to Antisocial personality disorder, because you hold it up as a specific example, and it's one of the most maligned mental illnesses... I'll just copy-paste:

A) A pervasive pattern of disregard for and violation of the rights of others, occurring since age 15 years, as indicated by three or more of the following:

    failure to conform to social norms with respect to lawful behaviors as indicated by repeatedly performing acts that are grounds for arrest;
    deception, as indicated by repeatedly lying, use of aliases, or conning others for personal profit or pleasure;
    impulsivity or failure to plan ahead;
    irritability and aggressiveness, as indicated by repeated physical fights or assaults;
    reckless disregard for safety of self or others;
    consistent irresponsibility, as indicated by repeated failure to sustain consistent work behavior or honor financial obligations;
    lack of remorse, as indicated by being indifferent to or rationalizing having hurt, mistreated, or stolen from another.

B) The individual is at least age 18 years.
C) There is evidence of conduct disorder with onset before age 15 years.
D) The occurrence of antisocial behavior is not exclusively during the course of schizophrenia or a manic episode. 

This diagnosis is a synthesis and outgrowth of conditions previously described as Psychopathic (same behaviors, attributing these to inborn qualities) and Sociopathic (same behaviors, attributed to socialization). Antisocial Personality Disorder eschews thoughts on origin and focuses on behavior and treatment. Behavior, you'll notice, corresponds very strongly with criminality. Criminality is key.

Earlier criteria dealt more with amorality or lack of a conscience, as do diagnostic methods (such as the Levenson Test). This is still a strong association, and we use Psychopath, Sociopath, and Antisocial Personality Disorder to describe people we feel lack a moral compass. But here's the thing- It's very possible to not know right from wrong inherently, and still be a descent human being by assessing cultural norms and making risk-benefit analysis. The people who can't tell right from wrong and are diagnosed Antisocial tend to be poor, victimized, and holding a criminal record.

There are people who suffer from APD and do not know it, and people who suffer from APD and are aware of it, and are not criminal. These people cannot discuss their condition, and are less likely to seek treatment, because it's associated with violence and criminality. Personality disorders tend to be egosyntonic, meaning sufferers are already less likely to seek help. Further stigmatizing the condition disinclines people to seek help.

So, yeah, I guess don't equate mental illness with bad behavior, or vice versa. If you don't think you do that, and someone says "please don't make that equivalency, it hurts me" listen to them instead of defending your right to make statements that hurt people.

I'm sorry this post was so long.