But False Hopelessness is way different from staying positive. I'll give you a scenario you are inside a burning house where you are near from the exit. Will you wait for the firefighters to save you or you can save yourself right away just by going out your front door? See that staying positive in the wrong scenario could place you in a wrong position and I don't think you want to be trap in a burning house waiting for help to arrive.
I don't find a lot of logic in your example and reasoning alike. In fact, I'm really curious if you actually don't understand my point. If you are caught in a burning house and you are hopeless, which means being in a state of utter despair - synonymous with being in the most negative state of mind ever possible, you won't be searching for an exit since it is all over for you from your own point of view. The whole idea of staying positive is that it gives you strength and power to look for a way out. That's why your example in real life would actually prove my point. In other words, no matter how bad your circumstances might be staying positive is the optimal strategy, if not overall the best one. Who has more chances to survive, the one who fights and hopes to pull through or the one who succumbs and gives up any hope?
It's not that hard to understand really.