It sounds like "it's complicated" is a better way of thinking about suicide prevention. It's pretty hard to measure a localized "it's no longer happening," when measures for suicide prevention are put in place.

Things effective one place, are less effective in others because of "sociocultural" reasons [0], but there is some evidence to suggest denying people access to the tools to commit suicide, and reducing media coverage (social contagion/clustering) seem to help.

With that said, yes, you are right, just changing one thing doesn't do much, there needs to be a bigger picture approach that addresses many factors.

[0] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1414695/