The excellent book "Paradoxes in Probability Theory" by William Eckhardt has some good arguments against this, the Simulation Hypothesis, and similar things. One simple way to summarize the counterargument is that in a lot of cases, the choice of seemingly-reasonable priors actually hides an unreasonable assumption that it is possible for the future to change the past.

Take a silly premise, get a silly argument and a really silly conclusion.

Confounding question: 1000 years ago, would this argument look any different? Answer: mathematically speaking, it would not. In fact, far more humans have been born than you could have predicted using this method. Conclusion: the argument is flawed.

The people living a 1000 years ago indeed could have used the same argument to show that they should be 95% certain to be in the last 95% of all humans to be ever born, and history indeed showed that this was not the case; the dice fell on the other 5% possibility for them and the population increased more than 20x.

However, the argument will still give the correct prediction for most humans that try to use it. Just not for the few that were in the special position to be born early in the sequence of all humans. The argument essentially tells you that you have no reason to believe that you are also in that special position.

10's of thousands of years of falling into the 5% group means either the dice are weighted or this isn't a good way to look at the problem.

You are trying to predict the length of the sequence of all humans, so you have no way to know whether or not you are ‘early in the sequence’. You can’t use the number you are trying to predict as an input to your prediction.

The tank problem doesn’t tell you how many tanks Germany will go on to build - it just tells you how many they have already built.

‘Good news! The war is almost over! The chances are these tank serial numbers all fall among the last 95% of the tanks Germany will ever produce!’

The difference is that the german tank sample can only be drawn from the pool of existing tanks so far. But when you consider your own birth, that's a sample out of the pool of all the humans, including future ones. Because future humans can also ponder their own place in the sequence, and where they find themselves is distributed uniformly over the sequence of all humans.

To apply the doomsday argument to tanks, we need to fix the sampling. It won't work for allies grabbing german tank samples. We have to think from the point of view of the tank; say a copy of your consciousness is being uploaded to an army of tanks. If you wake up as tank #734, could there eventually be millions of tanks? Maybe so, but there's only a 5% chance that you are one of the first 5% tanks, so there's a very good chance that there will be less than 734*20 tanks in total.