Oxygen and carbon are the most abundant elements in the Universe (formed via stellar nucleosynthesis), following only hydrogen and helium (formed via the Big Bang). There seems to be a lot of work to constrain the carbon-to-oxygen ratio [C/O] for low-metallicity stars, but I'm struggling to understand the motivation/reasoning. I think the idea is that very low metallicity stars that exist today should have enhanced carbon and/or oxygen abundances, otherwise they would've formed as Population III stars and died as supernovae long ago. Is that right? Then does this automatically imply that low metallicities should coincide with low [C/O] since oxygen is more abundant than carbon? Wouldn't this depend on the gas out of which a star formed and how many high-mass vs. low-mass stars chemically enriched that pre-existing gas?
Relatedly, what is the motivation for studying [C/O] vs. metallicity (e.g., as traced by oxygen abundance [O/H]) for planets and galaxies, and are the correlations expected to be similar (i.e., low overall metallicities coincide with low [C/O] ratios)?