I believe there are two types of suicidal people. Most of the people here seem to be suffering from long term issues and are very rational in their choices. I don't think there's much room for argument there, and if you have low quality of life for whatever reason, nobody but you should have a say. Noone else has to live with the pain.
But there are people who simply have a bad day and impulsively leap to suicide as the only solution. This is not rational, and I don't believe something this serious and permanent should be a first response. At the very least, the argument should be to wait awhile and make sure you can't live beyond it. If someone fails a test or breaks up with a person they've known for a week or gets a parking ticket, I think it's wrong to NOT argue a suicidal impulse.
If there's a fixable solution, I think that should be attempted first. Some things like hypothyroidism or low Vitamin D levels can make you suicidal, but fix the imbalance and the urges to die go away. It wasn't a true desire. With something like that, I think there's definitely a good argument against it.