If we expect a longer lifespan and higher odds of survival for potential children, our instinct is to delay reproduction and have fewer children in order to increase resources available per child.
This instinct developed over a very long period of time, but our technological advances have created a degree of safety and abundance that our instinct was never “trained” on and thus I suspect it is over-suppressing fertility. A longevity breakthrough will appeal to this instinct even more strongly than living in one of today’s highly developed countries. Consequently, I think our instinct will then suppress fertility even further.