Why isn't every finite locally free morphism etale?

Here is an example of a finite locally free morphism which is not etale: take spec of the natural inclusion $\Bbb F_2(t^2)\subset \Bbb F_2(t)$. This fails to be etale because it's a non-separable field extension.

The flaw in your reasoning is that your identification of $f_*\mathcal{O}_X$ as $A^d$ locally is only as a module - it needs to be as a ring to say anything interesting. (You should also note that even in the case of $\Bbb R\subset\Bbb C$, your idea is wrong - this is a point mapping to a point, while your reasoning would have two points mapping to one point.)