Where have I gone wrong in "proving" this big-O identity?
You should be able to work out the answer yourself.  Here's the general procedure:

Since you know of a counterexample to the original claim, plug those $f,g$ into each step of your proof and find the first step in your proof where the proof goes wrong.  In other words, find the first claim made in the proof that isn't correct for that choice of $f,g$.  That will help you pinpoint the step in your proof that is flawed, i.e., that makes a conclusion that doesn't follow logically from the previous steps.