Seems likely this is a gradient, not a binary thing, so perhaps someone with a beneficial mutation would just live a little longer. There’s not a ton of evolutionary pressure at the old age side of the spectrum, you’re not reproducing a lot, so it may not get picked up.
Wilder speculation: it could be that indeed this has been happening and is why humans live about 2x as long as chimps and gorillas.
Personally I think it’s quite unlikely that in humans single genes would have large deltas on age. There are maybe a few hints from supercentenarians (https://en.wikipedia.org/wiki/Research_into_centenarians) that telomerase and other genes have some effect, but the “one gene for aging” in humans is elusive. As a result, it’s hard to predict how all this would play out evolutionarily.