Hi Mark - I am an advocate for the idea that your life and your body belong to you. However, the statement, "I've never understood why suicide prevention is such a big issue. If it weren't, the stigma attached to suicide could be much reduced." doesn't reflect a few aspects.
1) Suicide is a permanent action. Many people are dealing with very temporary problems. This is perhaps the biggest reason suicide prevention is "such a big issue" - to prevent unretractable actions from being presented as solutions.
2) The approach that "a man...should be able to do so" is an individualistic idea - perhaps one that we could imagine in a non-collectivist Anglo-Western culture. But in some cultures, lives are seen as communal - as is the solution to problems in them. Or perhaps the public "investment" (not necessarily financial, though including finances) is great enough to need to avoid suicide.
Different cultures see this differently - and one is not "more right" than another. In smaller societies, such as those our minds/brains developed in, losing very many full-fledged adults to suicide would represent a major difficulty for the tribe.
3) Suicide often, if not usually, is strongly influenced by many societal factors. Different demographic characteristics predict differing levels of suicidality. If someone is assaulted, severely injured - any number of things, we need a societal approach to support people going through these issues -- and importantly -- to prevent recurrence, as having people who are victims of certain circumstances simply "exit" doesn't solve the root problem. Seeing suicide as something that "shouldn't be a problem" devalues the lives of certain people who have been subjected to certain experiences over the lives of others - by suggesting that suicide as a response to X issue - is not a problem, when actually, it can point to a massive issue in the society.
4) Lastly (part 1), and a key - is that the mental state of most people who approach suicide is not primarily determined by "stigma" as you suggest. Suicide is a feature in many societies, and in some, if not most, there are types of suicide (such as a "suicide mission") that are even lauded.
A huge problem with suicide is the mental state attached to it - one of deep suffering and inescapable angst. There are religious / spiritual traditions that consider the state of mind - rather than the action - to be the primary concern. So, suicide might not be the issue, but rather the "state of mind" at the time.
Yet across cultures with varying beliefs about the honour or lack thereof regarding suicide, where suicide can be attempted at different times and by different people with highly varying levels of stigma -- the majority of people who die by suicide -- regardless of the cultural approach -- do so with an unsettled, painful state of mind.
This may be in part because...
5) (part 2) There is a huge element to self-preservation in life. This is not "just" a human imagining - and it doesn't derive from "stigma". Choosing to end one's life generally is at odds with the entire flow of life -- thus the state of mind is rarely one of calm composure.
I welcome your comments. I request, however that you refrain from mentioning specific suicide methods/means in the future.