a) Insane people generally do not commit crimes and remain within the population.
Yes they do. The majority of mentally ill patients that commit crimes commit petty ones, or drug crimes, and not violent ones. The murderers and rapists and the violent offenders absolutely get institutionalized, if not imprisoned, but for the majority of the people we're talking about this just isn't the case.
b) Most insane people are in lunatic asylums.
This right here shows you have a fundamental misunderstanding of mental illnesses. The majority of people with bipolar and schizophrenia are not, in fact, in asylums. The majority of people with mental illnesses, like these, are living in the populace and using medications. Furthermore, many of the conditions I listed do not in fact make you insane 24/7. Bipolar people are only 'insane' when having manic episodes. Schizophrenic people typically present with both psychotic and non-psychotic phases.
People with endocrine disorders are not actually insane at all, but when left unchecked can lead to delirium. Too much thyroid hormone, or too little, can both cause this type of delirium, for instance.
c) The probability of having a mad man being off his medicine away from the asylum and coincidentally committing a crime is extremely low. Within a day his someone would report his strange behavior.
Actually, again, most of the mentally ill folks do not in fact live in asylums. And even if they did: Do you not see the problem with this logic? Within a day someone would report his behavior... And then he might be shot for running.
d) A mad man would find it very difficult to commit the crime since his thoughts are not coherent.
The lack of coherence is precisely why they might run from the police. I don't see how this supports your argument.
e) A mad man would not get very far running from the police. They would most likely already know about his madness beforehand and know how to deal with him.
No, a mad man wouldn't get very far running from the police if the police shot him in the back. There is no guarantee that they would know about his madness. Furthermore, his madness could be caused by something completely unrelated to mental disorders. It could be an issue with the endocrine system (diabetes, thyroid hormones). Short term disorders: Insomnia. Or other disorders which almost never classify a person as being 'mad' in and of themselves, but in fact can cause issues of madness due to side effects: Narcolepsy.
Lastly, you're not addressing one of the very major flaws in all of this logic: That mentally ill people are
not diagnosed at birth. In the vast majority of cases people find out they are mentally ill by having a mental breakdown. For bipolar folk this will typically present during the young teenage years, and most often will be diagnosed after a suicide attempt, or a manic episode.
If you have to first go through a mental breakdown to be labeled mentally ill, you could very well be subjected to being shot in the back, without even knowing what your own disorder is. You say this would be rare, but again, the rate of schizophrenia is widely accepted to be about 1 in 100. Bipolar disorder is recognized to be around 2%.