One flaw with the analogy is that a pretty significant portion of suicides (successful or attempts) are impulsive. Many people who attempt it have literally no idea they will when they wake up that morning.
A 2001 University of Houston study of survivors of near lethal attempts showed that 70% of them had only decided to do it an hour or less before the attempt, and 24 percent had only spent 5 minutes before the attempt.
If when attempting it, they discover that their chosen method won't work for some reason, many will just give up on killing themselves rather than looking for an alternative measure.
This has been reliably seen many times. If a change is made such that one suicide method becomes unfeasible, the overall attempt rate drops. If one were committed to offing oneself it is almost impossible to stop them, as there are countless ways to do it. But the statistics are very clear that if some method is removed, alternatives are not always used.
The British had an issue with people killing themselves with oven fumes, back when the ovens used coal gas which created a lot of carbon monoxide, with this accounting for about 50% of suicide deaths. When things were changed to use natural gas where this approach does not work, suicides dropped by 30%. Obviously while that means a portion ending up using some other method, many others simply never killed themselves.
Of 515 people prevented by police from jumping off the Golden Gate Bridge between 1937 and 1971, by the end of the 70s only 6% of them had committed suicide by other means.
Weirdly even some of the non-impulsive cases where people had planned an attempt for days or weeks just give up if their plan is foiled, rather than coming up with some alternative method.
Now obviously not everybody is like this. Some people will continue to try over and over until they succeed. But the numbers suggest that those are very much a minority.