Why aren't all isogenies over finite fields isomorphisms?
Solution 1:
An isogeny is surjective over the algebraic closure of $\mathbb{F}_p$. It is not necessarily surjective over $\mathbb{F}_p$ itself.
To give a simple example illustrating the issue: the map $f(x) = x^2$ is surjective over the algebraic closure of $\mathbb{F}_p$, but it is not surjective over $\mathbb{F}_p$ itself. In fact this example is more or less exactly what is going on in the isogeny setting.