The proof here has the following inequality:
But shouldn't it be the case that, by the Maximum Modulus Principle, $|f(z_r)|=1$, not just $\le 1$, since we are initially given that $|f(z)|\le 1$, and so it must achieve $1$ somewhere - namely, on the boundary of the disk. Moreover, how do we know that $f(z)$ is even defined on the boundary of the disk?
In addition, how do we know that $f'(0)\ne 0$?