LLMs make mistakes quite often. It is unreasonable to expect that it stops making mistakes just because suicide is mentioned.
LLM is a tool. There are many tasks it is good at and even more tasks where it sucks e.g., GPT-4 can fail on trivial chess questions.
Avoid permanent solutions to temporary problems.
> It didn't sound like he had a happy life to begin with
Which are exactly the kind of people who are going to be using LLMs as a "therapist". See the problem?