Proving a function is or is not a pseudorandom function F_k(x) = F_k(x)||0
You have shown a distinguisher for $F^1$ with high advantage that does not involve a distinguisher for $F$, so you can conclude there must be an error in your argument that that any distinguisher for $F^1$ implies a nearly-as-good distinguisher for $F$.

In particular, in equation $(*)$, you have confounded a uniform random function with the codomain of $F$ and a uniform random function with the codomain of $F^1$ by calling them both $f(\cdot)$.  It may be helpful to label the codomain sizes, or at least to call it $f$ vs. $f^1$, and clearly write out the distinction between the two and how your distinguisher acts on $f(x)\mathbin\| 0$, $f^1(x)$, and $F^1_k(x)$.