NB: based on some quick searches, it seems that low-metalicity Pop III stars would rely on the pp (proton-proton) fusion chain. That's going to slow reaction somewhat, and extend lifetime. But for high-mass stars with only a few millions of years expected lifetime in a Pop I/II class, that's ... still a relatively modest difference compared to the several hundred million year lifespan of the early Universe.

Or am I missing something?

Another interesting quirk of Pop III stars is that their initial mass function is expected to form much more massive stars than Pop II or Pop I. So even if Pop III stars are longer lived at the same mass as Pop II or Pop I, there will be a lot more supernovae per time, leading to fast enrichment and then Pop II.

That was one of my thoughts.

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.)

Thanks!