I am not a scientist either, but I share your skepticism.
All they can measure is how fast the population accelerated in modern times, when there is little to deter it. A bit of war, famine, plague, genocide, abortion, etc., in recent times.. other than that, mathematicians could measure the rate of population acceleration absent any extinction level event.
Then, divide 8 billion by that rate of acceleration, and calculate how long ago the number would be zero. That’s probably where they got the first near extinction ‘data’ point.
Beyond that, it would have to be pure conjecture.
A lot about past populations’ numbers can be estimated from genetic clues. (My Dad’s PhD was in Forest Genetics.)
That dip #3 is near certainty (under 1000 breeding individuals) and coincides right with the temperature pattern. Damn close call...