I am surprised that some smart people have been discussing the doomsday argument seriously for decades. The whole premise is that "supposing the humans alive today are in a random place in the whole human history timeline" assumes that we can apply the idea of probabilities and probability distributions to a one-off event. This particular piece of math works when applied to statistical inferences, "the process of drawing conclusions from data that are subject to random variation", not to a single data point with no way to repeat the experiment. There is no reason to think, or to check, that this particular map corresponds to this particular territory. To quote Maslow, "I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail,", and the doomsday argument is a good example of trying to hit a not-nail with a hammer.