"This is unlikely to be a convincing argument to a genuinely suicidal person."You may well be right. The problem is that suicidal people are usually depressed, and depression (as others have said [1]) is an illness. Specifically, a biochemical illness the affects the functioning of the brain to the extent that severely depressed and suicidal people often don't think rationally about the consequences of their actions. I think it is strongly arguable that trying to prevent the suicide of a mentally ill person is morally justified, even at the cost of temporarily impinging on their freedom of action.
I'm not saying that suicide is always irrational. Faced with, say, terminal illness and severe pain, suicide can be a rational choice for a person capable of rational thought. And I definitely support the right of an individual to end their life in that or a similar situation. Ultimately they "own" their life.
"what standing does some stranger on a crisis line have to try and do that?"
Maybe you haven't thought this through. When a person phones a crisis line they are actively seeking help, and have an implicit understanding that the person they speak to is there to try and provide it. "Standing" doesn't come into it because the caller chose to made the call and controls when to end it.
[1] https://news.ycombinator.com/item?id=5143839