I don't agree with him. The average lifespan for a human, barring incident or extreme illness, has been a steady 70-80 for some 10,000-100,000 years. Even with the leaps and bounds in medicine in the last 100 years or so, that hasn't changed.
But it has changed. It's risen dramatically across the world. Life expectancy for a man in the United States in 1915 was 53, for a woman 58. Today it's something like 76 for a man, 81 for a woman. (It's hard to be precise because of differing calculation methods.)
There are thirty countries where the life expectancy is eighty years or more today. Only in Sierra Leone is life expectancy as short as it was in the United States of 1900.
You might argue that the oldest human beings can expect to get hasn't changed much in the past century, but I'm not even sure of that. The number of centenarians has been rising quite noticeably in recent decades.
Is that because people are eating better and not dying in famines, plagues, and wars as much as before? Quite likely. Because we're better at taking care of people? Also quite likely. And we're better at taking care of people because society knows more, and is wealthier, and more strongly feels that individual health is a positive public good, too. Those seem to me trends likely to continue in our world and, certainly, in Trek's timeline.