quantum mechanics - Physical intuition behind why $hat{J}^2$ and $hat{J}_z$ commute

In Townsend's Quantum Mechanics textbook, he shows that $\hat{J}^2$, the squared magnitude of the angular momentum, and $\hat{J}_z$, the generator of rotations about the $z$-axis should commute. I understand the proof, but I would like some help understanding why, from an an intuitive point of view, these operators should commute.

At the end of the chapter, Townsend writes that it makes sense that they commute because the magnitude squared of the angular momentum vector (i.e. $\hat{J}^2$) is not affected by rotations. Could you elaborate on this? Why does this mean that we can simultaneously measure the angular momentum squared and one component of angular momentum?