Rule with independent random variables and conditional expectations
The faulty step is when you assert that $E(X)E(Y1_A)=E(XY1_A)$. Here one must not only assume that $X$ is independent on $Y$ but that $X$ is independent on $(Y,Z)$.
To see that there is a difference, the usual example works here: take $X$ and $Y$ i.i.d. centered Bernoulli random variables and $Z=XY$. Then $X$ and $Y$ are independent (by definition), as are $Y$ and $Z$, as are $Z$ and $X$ (easy to check), but $X$, $Y$ and $Z$ are not*. And $E(XY|Z)=Z$ although $E(X)E(Y|Z)=0$.
*For example, $P(X=1,Y=1,Z=-1)=0$ but $P(X=1)P(Y=1)P(Z=-1)=1/8$.