I failed to find anything in the video where the lip-smacker says the Z carries weak hypercharge. The Z does not carry weak hypercharge, or charge, just as the photon (it's group theoretic brother) doesn't.
This is evident from weak mixing, since they, γ & Z, are a rearrangement of the weak hypercharge gauge field, B, and the $T_3$ one, $W^0$. Since these operators commute, the respective gauge fields are neutral w.r.t. them, and the eigenvalue of $W^0$ under $T_3$ is 0, while B is Abelian, so neutral, just as the phonon is such, under charge. So any combination thereof, like γ & Z, also has to be neutral!
Of course, γ & Z do couple to Q and Q & Y respectively, as this is their function in the model.