As you stated, it is incorrect. A set of conditions which imply this asymptotic is given by Theorem VII.6, which you mention in your remark. (The equation they consider is not F(x,y)=0 with positive coefficients, but y=F(x,y) with positive coefficients, and this is not the same!)

So what is your question, again ?

A set of conditions which imply this asymptotic is given by Theorem VII.6, which you mention in your remark. These additional condition is that the equation is "a-proper". The theorem is not true without this additional condition.