Assuming that the number \(Y\) of all people who have ever been born and will be born at some point is limited, let \(x\) be your absolute position from the beginning of the list. Then \(0 < \frac{x}{Y} \leq 1\). We can now say with a probability of \(95\%\) that you are among the last \(95\%\) of all people ever born, i.e. \(0.05 < \frac{x}{Y} \leq 1\) and thus \(Y < \frac{x}{0.05} = \frac{100 \cdot x}{5} = 20 \cdot x\).
According to estimates, \(x \approx 6 \cdot 10^{10}\) and thus \(Y < 120 \cdot 10^{10}\). If life expectancy remains constant and the number of people living at the same time stabilizes, \(Y-x = 114 \cdot 10^{10}\) still has about \(10,000\) years left. The doomsday argument is equally valid at all points in history - one could make the same argument \(2000\) years ago or \(5000\) years in the future; the basic logic would still apply (the upper limit of \(Y\) becomes correspondingly larger).
The following thought experiment works similarly: Consider the two urns \(A\) with \(100\) balls and \(B\) with \(100\) million balls. You don't know which urn is which. If you now blindly draw from one of the two urns and catch a ball with the number \(42\), it is more likely to come from urn \(A\) than from urn \(B\) (it is also very likely that you are among the last \(95\%\) of all people ever born and very unlikely that you are among the first \(5\%\) of all people ever born).
So the urn is constantly filling up with new balls over time and pulling out a number at any point in time tells us something about the possible total number of balls at that time, but not something about the future number of balls in the urn. This would require an analysis of the urn.