If $K/F$ is a finite extension and $E$ is an intermediate field, then I can see that $K/F$ being simple implies $K/E$ and $E/F$ are simple since $K/F$ has a finite number of intermediate fields. Therefore, each of $K/E$ and $E/F$ must have a finite number of intermediate fields. However, why does this logic not work in the converse direction?
If $K/F$ is a finite extension and $E$ is an intermediate field, then how come $K/E$ and $E/F$ being simple extensions doesn't imply $K/F$ is simple? There are only finitely many intermediate fields in total, right?
If $ K/E$ and $E/F$ have only finitely many intermediate fields, this does not imply that $K/F$ has only finitely many intermediate fields. Surely, there are only finitely many intermediate fields contained in $E$, and only finitely many intermediate fields containing $E$, but this is not enough, since there are intermediate fields $L$ (possibly infinitely many) which neither contain $E$ nor are contained in $E$.
$$\begin{array}{cc} K & \\ \uparrow & \nwarrow \\ E &~~~~~~~ L\\ \uparrow & \nearrow \\ F & \end{array}$$
Consider the standard example $K = \mathbb{F}_p(X,Y)$ with $E = \mathbb{F}_p(X^p,Y)$ and $F = \mathbb{F}_p(X^p,Y^p)$. Then $K/E$ and $E/F$ are simple, but $K/F$ is not. It has infinitely many intermediate fields (SE/2944053).