The initial assumption is the Copernican principle, and I wouldn’t call it vacuous. It’s the Copernican principle that suggests the probability distribution – indeed, suggests the existence of a probability distribution. The statement that “any randomly selected value is 95% likely to be in the final 95% of the range” is trivial, but only once you know the distribution.

Also, let me make a subtle clarification: there is a 95% chance that you were born in the final 95% of all people. Your birth is a definable event, as opposed to “now”, which as you say is slippery and probably not amenable to statistical analysis. As in your analogies, there is no particular probability attached to simply existing among the final 95% of all people. It’s not obvious what that would actually mean.

However, given that your birth was a random selection from said probability distribution, we can estimate the distribution’s parameter. It’s a ghastly, piss poor estimate, granted. It could easily be an order of magnitude off, but it is an estimate nonetheless. We have one degree of freedom – the bare minimum for estimating one parameter. Thus, I wouldn’t characterise the doomsday argument as a prediction, but more as a kind of fuzzy constraint. You’re right that no self-respecting scientist would accept such pitiful data for any practical purpose, but the context here is rather special. This is all the data we’re going to get, and the problem is interesting enough that we may as well use it.

The argument is actually a lot more generous than I originally believed it to be, but I remain unconvinced. The argument’s initial assumption is vacuous (i.e. devoid of significance). To say that there is a 95% probability that we are in the last 95% of any uniform interval is mathematically trivial, but without knowing anything about the total population we have no way of gauging what fraction we are through that population. For example at any given time you could make the claim that there is a 95% probability that you are within the last 95% of that hour, but that does not tell you what the time is. Similarly, if you are in a running race (which you don’t know the length of), it’s perfectly valid to say that there is a 95% chance that you are in the last 95% of the race – but this still doesn’t tell you how far you have to go.

Probability theory is at its most useful when making predictions about a well-defined system. The result you get is highly dependent on the quality of information you input. A good way to make many physicists roll their eyes is to mention the Drake Equation. By the time you have entered all the variables (which includes numbers we have no reliable information on), the output has so much uncertainty that it is essentially meaningless.

Although it it allows for an interesting philosophical debate (providing my brain with a few hours worth of exercise), I believe that the system we are studying (human population past, present and future) is simply too complex to make any kind of reliable prediction based on such simple input.