Exactly, well said. Most, if not all of the stigma around suicide is because society has decided that it is immoral and unacceptable. When people say "why didn't they open up" well the answer to that is "because society has either turned a blind eye to someone suffering until it's too late" or "they wouldn't listen to them without judgment and an open mind, but instead threaten them with involuntarily commitment (being held against one's will)." Personally, I think if there has been less of that, then there will be more people willing to open up and be honest.
As far as demand increasing initially, yes, the initial wave of people would be the ones who wanted to die but haven't been able to die peacefully and legally (without being locked up against their will for planning or even attempting). However, when there are mostly happy people left and very few sad people and with the population dwindling significantly (e.g. we are at almost 8 billion people in the world, and then imagine that number dropping to about 1 billion). Thus, as a result of that, there will be more incentive to treat people better, more resources for the remaining humans around, and of course less stigma overall.