The $\theta$-angle poses a naturalness problem in the context of the Standard Model (SM).
In fact, as you correctly remark, when you consider QCD in isolation, sending $\theta \to 0$ restores $CP$-symmetry; that is, a small $\theta$ is technically natural according to 't Hooft. Much more strongly, if QCD was the whole thing, the experimental bounds on $\theta$ would simply suggest $\theta \equiv 0$.
However, in the SM there are other potential sources of CP violation: sending $\theta \to 0$ alone is not enough in order to restore CP; in order to restore CP, you must also to send to zero the CP violating phases in the CKM matrix.
These phases are, however, experimentally known to be non zero, which means that (from a naturalness point of view) you would expect an equally big $\theta$ angle.
This is why the strong experimental bounds on $\theta$ pose a naturalness problem within the SM.