Averaging more than 2 children attempts to grow the population at an exponential rate. This attempt fails because we are always at the limit. We are generally always at the limit because the exponential growth happens very fast relative to how long humans have existed in each separate environment. When a population is at a stable limit, the child mortality rate is forced to rise such that the extras above an average of two will not live to be an adult. For example, if we average three children, then one in three children will not survive to adulthood. If we average 2.5, then one in five children will die. The formula for the child mortality rate is (x-2)/x where x is the number of children we average.
This simple formula shows that we determine the death rate of children by how many we average. Unfortunately this formula is relatively unknown because for the past several hundred years we have increased the limits at exponential rates. This has fooled many into believing that we have never been at the limits. When the limit increases, the child mortality rate will be lower and the connection to the birth rate is not obvious.
Even though we have managed to expand the limits by making fertilizers, better crop varieties, refrigeration, and transportation to name just a few, we have not expanded it as fast as our births have demanded. When a population is at the limit the population experiences pockets of high child mortality with low adult life expectancy and these symptoms have always been present in the world human population.