It isn't true by definition: you could quite happily induce people to release a series of highly contagious bioweapons, after which those people would be surplus to requirements. What is true is that you're likely to need humans to sustain you and act for you for a few years to decades, so if you're not suicidal or deeply mad (and that is itself by no means self-evident) then total and immediate human extinction is probably not something you will aim for. But ruling out total, prompt human extinction isn't, by itself, remotely enough to justify the OP's overall don't-worry conclusion.

(Again, to be clear, I myself am not predicting or assigning a significant probability to any doom scenarios, because I do not expect AGI.)