Many thanks to Paul Almond for developing the initial form of this argument.

My previous post was somewhat confusing and potentially misleading (and the idea hadn't fully gelled in my mind). But here is a much easier way of seeing what the SIA doomsday really is.

Imagine if your parents had rolled a dice to decide how many children to have. Knowing only this, SIA implies that the dice was more likely to have been a "6" that a "1" (because there is a higher chance of you existing in that case). But, now following the family tradition, you decide to roll a dice for your children. SIA now has no impact: the dice is equally likely to be any number. So SIA predicts high numbers in the past, and no preferences for the future.

This can be generalised into an SIA "doomsday":

Everything else being equal, SIA implies that the population growth rate in your past is likely to be higher than the rate in the future; i.e. it predicts an observed decline, not in population, but in population growth rates.