How to remove null production from context free grammar?
Your simplification is wrong. The language of the first grammar includes $b$, but the one of the second grammar does not.
If you replace the definition of $A$ in the productions of $B$ you obtain $B \to b \mid \epsilon$. Then if you replace $A$ and $B$ with their definition in $S$ you have.
$$
S \to a \mid bb \mid b \mid aba \mid aa
$$