Death from diabetes or cancer in old age does nothing to prevent reproduction. Arthritis, tooth loss, obesity, heart attacks... none of it.
A human only has to survive for ten years or so to reproduce, given adequate nutrition, or around fourteen years without... and just twenty to thirty years to both reproduce and support their children till they can forage for themselves.
You're all thinking in terms of large populations. Humanity wasn't always a large population. There's evidence of an evolutionary turning point wherein the entire human race was down to a few thousand people. This is the turning point at which we probably became what we are now.
After that turning point, there have been other occassions in which mutations could be selected and spread, even when the human race exceeded a million heads.
Scattered tribes of humans, in groups of dozens and hundreds, or at most, thousands, have lived separated from each other for tens of thousands of years before the big civilizations arose. And in these scattered tribes, we see some of the more modern evolutionary adaptations... like the beneficial mutation of lactose-tolerance, which allowed certain tribes to drink animal-sourced milk. Which allowed them to travel further and conquer other territories. That beneficial mutation has affected much of human history, actually..