The thing is, escaping predators, eliminating competition and securing resources were being carried out alongside having sex and reproduction. So no matter how they fail at the former, the latter was always there to keep things going.
Your logic is correct, though, you have to survive to a certain age or for a while first before you can reproduce so staying alive was a priority. But something being a priority doesn't mean what comes after (the secondary consequence) can not play a bigger role than the first.
When we say the early man was able to stay alive till they gave birth to children it might seem like the early man was such an expert at staying alive but he might have been terrible at it, only making the bare minimum score, and it wouldn't matter because sex and reproduction is so easy and in geometric progression.
The requirement of staying alive was fulfilled even if one only barely managed to stay alive. One didn't have to stay alive for so long in order to reproduce. Everyone could have died in their twenties and it wouldn't matter. It's just like a man telling his son that if he makes the minimum grades he would make him CEO at the company then his son makes minimum grades and becomes CEO. Without him making minimum grades would he become CEO? But that doesn't mean making minimum grades plays a bigger role in the equation than his dad.
That's how it was for the early man. Sex and reproduction told the early man that all he has to do is live long enough to birth and it would make sure his specie never goes extinct. Staying alive is commendable, making minimum grades is commendable, I'll never down play them but what came after was just deus ex machina! An insane game changer.