The significance of die out (as opposed to simple die) can be seen in this Cambridge Dictionary definition...
die out - to become less common and finally stop existing (italics mine)
As with peter out (to gradually stop or disappear) and wear out (to become useless from long or excessive wear or use), the preposition primarily connotes the extended and continuous process, rather than the eventual end. It's worth noting that all these phrasal verbs often occur as continuous forms (dying out, petering out, wearing out), reflecting that connotation.
In the real world it's often not easy to say with certainty exactly when the relevant process has actually finished, which is why it's quite natural to say things like The species died out completely, This shirt is completely worn out when we want to emphasise that the (preceding) drawn-out process has reached its eventual conclusion.
In short, I don't sense any real tautology in such usages. In many contexts, the "extraneous" preposition out actually provides a useful way of being more precise in expressing exactly what we want to convey, but sometimes it's useful to include completely to ensure that the ongoing process connotations don't overshadow an intended reference to the end result.
Consider this written instance of...
1: The Camondo family died out
...and contrast it with the possible alternatives...
2: The Camondo family died
3: The Camondo family died out completely
...where he actual wording (#1) is both less "sudden" than #2 (perhaps they all died at the same time) and less "absolute" than #3 (it's quite certain there were no survivors).