I'm not up on the literature and haven't thought too hard about it, but I'm currently very much inclined to not accept the premise that I should expect myself to be a randomly-chosen person or person-moment in any meaningful sense—as if I started out as a soul hanging out in heaven, then flew down to Earth and landed in a random body, like in that Pixar movie.

I think that "I" am the thought processes going on in a particular brain in a particular body at a particular time—the reference class is not "observers" or "observer-moments" or anything like that, I'm in a reference class of one.

The idea that "I could have been born a different person" strikes me as just as nonsensical as the idea "I could have been a rock". Sure, I'm happy to think "I could have been born a different person" sometimes—it's a nice intuitive poetic prod to be empathetic and altruistic and grateful for my privileges and all that—but I don't treat it as a literally true statement that can ground philosophical reasoning. Again, I'm open to being convinced, but that's where I'm at right now.

Related LessWrong discussion / paper

With low confidence, I think I agree with this framing.

If correct, then I think the point is that seeing us at an 'early point in history' updates us against a big future, but the fact we exist at all updates in favour of a big future, and these cancel out.

You wake up in a mysterious box, and hear the booming voice of God:

“I just flipped a coin. If it came up heads, I made ten boxes, labeled 1 through 10 — each of which has a human in it.

If it came up tails, I made ten billion boxes, labeled 1 through 10 billion — also with one human in each box.

To get into heaven, you have to answer this correctly: Which way did the coin land?”

You think briefly, and decide you should bet your eternal soul on tails. The fact that you woke up at all seems like pretty good evidence that you’re in the big world — if the coin landed tails, way more people should be having an experience just like yours.

But then you get up, walk outside, and look at the number on your box.

‘3’. Huh. Now you don’t know what to believe.

If God made 10 billion boxes, surely it’s much more likely that you would have seen a number like 7,346,678,928?

1. The distinction between humans and other lifeforms is arbitrary.
2. ~10^30 lifeforms have ever lived.
3. Thus, there'll likely be another ~10^30 lifeforms.
4. But it's equally likely that this will be:
   1. ~10^30 non-human animals, if we do go extinct (life originated ~5bn years ago and life will end in ~5bn years when the sun dies).
   2. ~10^30 (post-) humans, if we don't go extinct (as per Bostrom's calculation).
5. And so the doomsday argument doesn't tell us anything about x-risk or the far future.

As I see, the point is to estimate when extinction would occur by estimating the distribution of population accross time, right?  So we use a Rule of Succession-like reasoning... I'm ok with that, so far. N humans have lived, so we can expect more N humans to live, we can update our estimate each time a new one is born...
But then, why don't we use the time humnas have already lived on Earth as input instead? I mean, that's Toby Ord's Precipice argument, right? So 200k years without extinction lead you to a very different guesstimate. 

One reason I'm not convinced by  the Doomsday argument is that it's equally true at all points in history - you could make the same argument 2,000 years ago to the Greeks or 10,000 years into the future (well, only if Doomsday isn't really imminent) and the basic logic would still hold. I find it hard to be convinced by an argument that will always come to the same conclusion at any point in history, even though the argument is that we're most likely to exist at the point that it's true.

The problem with the analogy is that the urn is continuously filling with balls with higher and higher numbers, so pulling out one number at any point in the process tells you nothing about the future number of balls in the urn. That would require analysis of the urn and the ball-dropping mechanism.

For this reason, I find concrete existential risks much more convincing than the Doomsday argument.