Can spurious correlations exist in the (theoretical) population? Is it possible that spurious correlations exist between two random variables X and Y in the (theoretical) population (here I mean purely by chance, not due to missing confounders in a model, etc.)?

As suggested in Ben's answer in the thread "Spurious relationships: flavours, terminology", the qualifier spurious can be attributed to our interpretation of some statistical finding. We may have a nonzero correlation between $X$ and $Y$ in a population (or determined by a data generating process) where neither $X$ causes $Y$ nor $Y$ causes $X$. (Monthly ice-cream sales on a beach and drownings on a neighboring beach would be a good example. Both vary seasonally due to weather conditions and working vs. holiday periods, so they will be positively correlated, yet none causes the other.) Interpreting this correlation as if $X$ causes $Y$ or the other way around would be a spurious interpretation.
(Focusing on a population instead of a sample rules out the case where a correlation is estimated to be nonzero while it is actually zero in population / by a data generating process. Some practitioners would call such a finding a spurious correlation.)
