Pop III stars (if they existed) are really a mystery, we can't easily extrapolate. These stars would be purely hydrogen and helium so it would take them a surprisingly long time to get to CNO cycle, for example.
Or am I missing something?
Another is that p-p fusion is fairly common. Look up when the Big Room's bright and you'll see a ... stellar example yourself ;-)
(p-p fusion dominates in stars < ~1.3 M, where M is a solar mass. CNO fusion is typical of more massive stars.)