I agree it's a flawed argument. The base concept comes from Bayes' theorem, to the effect that if you randomly choose a member of a set, the characteristics of that member are most likely to be the most common characteristics found in the set as a whole. In other words, if you have a bag full of red and blue balls, and the ball you randomly pick out is red, it's (non-conclusive, statistical) evidence towards the conclusion that there are more red balls than blue balls in the bag. That's fairly uncontroversial, although it's important to note that it's a statistical probability, not an entailment. Your random draw could always be that one-in-a-million long shot, although probably only once in a million tries. (More about this here.)
A more controversial application of this theorem is the idea that you can use yourself in place of the randomly chosen ball and extrapolate info about all humanity from your own personal traits. Personally, I think the argument breaks down right here. One of the major, obvious difficulties, is that you aren't necessarily a random, representative member of humanity. Most people in the world right now live in poverty. Most people in history didn't have access to the internet. The fact that you've even heard of the doomsday theory and are considering it may already entail that you are a special person with non-representative traits.
Another objection against the doomsday theory is that it's simply mathematically unsound. It stems from the idea that there are more people alive now than have ever been alive at one time in the past. Imagine a graph of people throughout all time. A basic assumption of the theory is that population rises to a maximum, and then doomsday, the graph cuts off. All other things being equal, you (supposedly!) as a random datapoint, should statistically be found right at the righthand edge of the graph, where the population is highest.
But that misunderstands the math. This is the moment in time when the population is the highest it has ever been. But there are more people who have lived and died than are alive right now. So a random datapoint is likely to be on the righthand side of the graph but NOT necessarily all the way to the edge. We also have no way of knowing what the full shape of the graph would be --why must it come to a peak and then cut off? Couldn't it gradually diminish again? In that case the most likely place to find a random datapoint would definitely not be the end.
With all that said, I think your objection fails against this argument, because the putative relationship isn't causal at all, it's purely statistical. And the piece that you're missing is that there isn't a constant number of people born each year --there are more people born each year. So the probability that the event "someone comes up with the DA this year" happens is bigger each year --all other things being equal, and as long as population continues to increase.