Ah, but if we sample at a random point in time with respect to an exponentially growing population (like lemmings or humans), then we are more than 5% likely to sample from the initial 5% of lives, simply because those lives are spread over a long time period.
We therefore can't move on to state that the birth number we find on sampling has only a 5% chance of existing in the first 5% of lives, and we've crippled Carter's reasoning. In fact, if we sample at a random time, we can't deduce a 95% confidence interval without knowledge of the shape of the population curve ... and if we have that, Carter is superfluous.

The elegance of Carter's insight is that sampling a random life (simply by living), allows you to forget the shape of the curve and derive a confidence interval with no other information but your birth number. Random sampling in time will only give equivalent information if lives are uniformly distributed along the time axis.

Grant Hutchison