If you do this calculation today, then you expected humanity to exist untill X

But if a hypothetical medieval person did the same calculation based on data avaliable to him, then his answer is Y

For a hypothetical caveman it's Z, where Z is rather small.

Returning to today, X grows every day.

These results are non-sensical, seems like the base assumption is invalid: "it says that supposing that all humans are born in a random order, chances are that any one human is born roughly in the middle"

Actually a medieval person or neolithic person would have derived large future ages, because in those epochs population growth was slow and arguably not even exponential.

The fact we know there is a fast rise and possible peak in world population around 2050 means most humans born or ever to be born will be born in the years ~1700-2200. This is pure math, and an argument against it is an argument against the Copernican principle.

Yup, i got it wrong. However it seems like a statistical argument from a sample of 1, very similar to this:

https://xkcd.com/605/

Could method adequately predict extincrion of DoDo or brontasaurus?

Why do human ancestors stretching back ~1B years not count?

They could, but most humans ever born have been born in the last 300 years.

You're missing the point, though. Count pre-human ancestors and that's no longer true.

It ceases to become a "doomsday" argument and more the (fairly obvious) idea that our species is likely to continue evolving to some extent, possibly to the point where we're something posthuman.

Most humans ever born so far.