This seems like a straightforward sort of case that philosophers would normally use as an argument against the frequency interpretation of probability. Look at page 14. The two assumptions highlighted there are ones that philosophers of probability generally already think are false. It can be useful to model probabilities with frequencies in some actual finite system, but there's no good reason to think that's just what "probability" means, in the sense that we've always been using the word. There is no actual physical event that corresponds to the "end of time" that they talk about.

And in fact, if you do the limiting procedure in the relevant way, then you won't get their result. Sure, if you take the limit by imposing a cutoff that has some geometric shape to it, and then expanding that shape, then you get their result. But if you take the limit by starting with a geometric cutoff, and then extending to the end of the life of any observers that are even partially included, then you get the result that no observer reaches the end of time.

Also, this main point doesn't really have anything to do with Bostrom's Doomsday Argument.

The passage you quote though does seem relevant. Just because all observers eventually wake doesn't mean that the fact of waking can't affect your probabilities. Think about "sleeping beauty" cases for an example where all observers wake, and yet it seems reasonable to say that waking actually gives beauty evidence that changes her beliefs about heads and tails.