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.
So I think it is fair to say they did exist. If we believe in Big Bang Nucleosynthesis then heavy elements had to come from somewhere making the first generation of stars (whatever their properties may be) be Population III. I agree that without a catalyst it's hard to initiate the CNO cycle but indeed models predict that it is possible even under these circumstances.
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.
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.