As the population declines we would probably reach a point where we have to go back to agrarian societies and those pretty much need a bunch of child workers working the farm with the parents.
My thought would be no, we won’t disappear unless it’s a cataclysmic event that just wipes us all out at once. Also if we can get off this rock and establish a base somewhere else like the moon or mars that increases our chances of not being extinguished even higher.