I think it's precisely this. We see exponential population growth of population, so we naturally assume most humans are alive now. What this intuition fails to grasp is quite how long humans have been around: eventually that adds up.

With a precise exponential the tail wouldn't really add up all that much. The integral to infinity is a finite number.

True. Maybe the fallacy is seeing exponential(ish) growth now, and pretending it was always exponential. Maybe population was linear(ish), for a very, very, long time.

It can't be exponential. If in 1800 there was 1G people, in 1900 2G people and in 2000 6G people, and we fit an exponential curve thought that we get less than 10 people alive in the year 0.