Confusion with understanding the fixed bias circuit
The Ebers-Moll equation does account for changes in \$\beta\$, but they use a parameter \$\alpha\$ where

$$\alpha = 1 - \frac{1}{\beta} $$

We often say that \$V_{BE}\$ is fixed in this circuit because the changes in \$V_{BE}\$ are usually small compared to \$V_{CC}\$. It follows that \$I_B\$ is (essentially) fixed since it is determined by Ohm's Law and the voltage drop across RB.

Your understanding is correct then that if \$\beta\$ (or \$\alpha\$) changes then \$I_C\$ will change significantly, causing \$V_{CE}\$ to change proportionally.