I'm busy self-studying Schoof's algorithm from Andrew Sutherland's notes.
In section 9.6, he states that when we happen to find some factor $g$ of the division polynomial $\psi_\ell$, then the roots of $g$ must be the $x$-coordinates of points in the kernel of some endomorphism $\alpha$.
How could we justify that such an endomorphism exists, and can we say what $\alpha$ would look like (besides simply having a factor $1 / g$ in one of its components)?
Answering my own question.
Turns out the $\alpha$ referred to in the notes is exactly the endomorphism which we are trying to reduce mod $\psi_\ell$, although it isn't made very clear.
Explanation: Let $\alpha = (\alpha_x(x), \alpha_y(x) y)$ be an endomorphism in $\text{End}(E)$, where $\alpha_x$ and $\alpha_y$ are rational functions. If the denominator of either $\alpha_x$ or $\alpha_y$ has a non-trivial common factor $g$ with $\psi_\ell$, this means that there is some point $P \in E[\ell]$ for which $\alpha(P) = 0$. Why? Well, writing $\alpha$ in its projective form would allow us to clear denominators, thus leaving the $Z$-component of $\alpha$ with the factor $g$, so that we would necessarily get $\alpha(P) = (0 : 1 : 0)$.