I don't feel it would be appropriate to put too much of my own opinion into this discussion (because I happen to be very strongly pro-choice on the matter, having made several attempts including a fair number of Intensive Care Unit hospitalisations myself), but I think it would be appropriate to clear up a particular common misconception which it seems you are currently accepting. Namely, "Suicidal ideation can usually be treated in clinical settings."

In fact, the literature to date has suggested that modern clinical treatment markedly increases the risk of subsequent attempts. Specifically,

• On a population level, the per capita spending on mental health care, the number of hospital psychiatric beds available per capita, etc. have been directly correlated with suicide attempt rate in a population. Moreover, as spending and bed availability increase in the same country, so do attempt rates.
• On an individual level, patients who receive more treatment paradoxically become more likely to kill themselves the more treatment they receive. Some (many) have suggested this is because more severely ill patients are the ones receiving more treatment, which does not explain why patients who were not initially suicidal also often attempt suicide following major contact with clinical treatment; e.g., it is quite common for a patient hospitalised for a psychotic episode with no known history of suicidal ideation or behaviour to begin making attempts shortly following the hospitalisation.

Having been hospitalised for attempts myself and exposed far too often to the modern clinical system, combined with the best evidence available, I am of the opinion that modern treatment, if anything, increases both the level of suffering in the general population, and the risk of suicide itself.

If I had to propose an explanation for this phenomenon, I would say it's likely related to the "...wealth of academic and anecdotal evidence" which "documents how suicides traumatise witnesses (e.g. train drivers, police, paramedics, family and friends)." That is, said trauma is indicative of a self-focused response termed in the psychological profession as "personal distress," which has been shown repeatedly to sharply reduce empathy and lead to poor treatment of the perceived source of the trauma. The perceived source of the trauma is, of course, the person who is suicidal; the result is quickly and continuously escalating negative outcomes for everyone involved.