Chronotherapy offers an exciting possibility for improving cancer treatments by leveraging the influence of the circadian clock on the cell cycle. While several molecular interactions coupling the two oscillators have been identified, whether they lead to emergent control of cellular proliferation remains unclear. Using stochastic simulations, we demonstrate that the established gene networks underlying the two oscillators are sufficient to generate lineage correlations in cell cycle times, as observed in single-cell microscopy data. The interactions also create a 'therapeutic window' between cancer and normal cell proliferation peaks that can be leveraged for chronotherapy. Surprisingly, our model predicts that KL001, a clock inhibitor, minimally affects population growth but significantly alters lineage correlations. Our results suggest that clock control of the cell cycle may not be detectable by measuring changes in population dynamics, but combining measurements of lineage correlations with KL001 treatment may provide a more sensitive approach to detecting the coupling.