Lathes can make cylinders, but not of unlimited length in one setup so they lose some accuracy making cylinders longer than their carriage travel. And their beds are by necessity longer than their carriage travel, since the carriage rides along the bed and isn't infinitely thin. They also can't make things like motor stator laminations, and you definitely need a motor for a replica of a motorized lathe. So lathes can't replicate themselves exactly.

Milling machines are also just lathes with a different orientation, an extra travel axis, and a motor optimized for higher speeds & lower torques, it's possible (and reasonably common) to use a mill like a lathe or a lathe like a mill in many cases. So "only machine" part is also a stretch.

if the purpose is to bootstrap, you could also use something like a leather belt drive off a central shaft, which would require different power sources but ones that a higher percentage of could be made with a lathe

> Milling machines are also just lathes

Mills are more limited than lathe - they don't have leadscrews, which are a necessity in the "build yourself" phase. You cannot make arbitrary threads with a milling machine. Thread milling gives some capability in this arena, but that's a CNC process.