"there were lots of other people around at that time whose lineages gradually died out"
It seems to me that if those lineages aren't in the DNA data, then they didn't --in the intervening years-- intermix with the lineages in the DNA data.
So, ipso facto, those lineages went extinct, and the 1280 ancestors posited in the study are the non-extinct lineages. QED.
Now, that doesn't necessarily mean they were isolated from each other, however. If we suppose that the population was actually 5000 people (just an example; I have no idea if this is a realistic number), then some percentage of those people died with having children, some may have seen their children die, etc. Of the 1280 people who still have living descendants, many of them would have had children with those people who did not, but those children (or grandchildren, etc.) eventually did not have any children themselves.
Within your own social circle, you likely know some people who will still have living descendant generations from now and some who will not.
If those lines died out from intertribal competition though, it wasn’t a near-extinction era. Some slightly less competitive group could have made it into the present day in that case.
It seems to me that if those lineages aren't in the DNA data, then they didn't --in the intervening years-- intermix with the lineages in the DNA data.
So, ipso facto, those lineages went extinct, and the 1280 ancestors posited in the study are the non-extinct lineages. QED.