The opposite is in fact true, hunter gatherers were short lived and agriculture is what gave humans the first great advance in longevity.
https://en.m.wikipedia.org/wiki/Life_expectancy
That said, diet probably isn't as crucial to longevity as we'd like to think. Hunter/gatherer populations got (and get) a lot of exercise and lead relatively low-stress lives. For example, the Tsimane of Bolivia eat animals and starch and have practically no heart disease. They walk around a lot and are generally a happy bunch. Probably the only thing that would increase life expectancies more would be actual medical care, particularly regarding parasites.
In context of ancient world, nutrition of the family also had other mechanisms that impact the survival of children (strength to build housing, or strength to defend against hostile people, etc).
That was the "advent of agriculture", also known as "Neolithic revolution".
However, overall I'd say that starvation vs. having sufficient food impacts mortality rates more than modern medicine. You can't eat the best of hygiene and medication doesn't nourish you.