Question about the definition of a secure PRF
You have the math right, but you seem to have mis-interpreted the formulas.  So, let me try to walk you through it.

The "advantage" of an attack is the difference $|\Pr[Exp(0)=1] - \Pr[Exp(1)=1]|$.
The advantage is a measure of how effective the attack is.  If the advantage is large (significantly greater than 0), the attack is successful (and the function is not a secure PRF). If the advantage is negligible, the attack is ineffective.

You suggested we look at an attacker whose algorithm always returns one.  In this case it turns out that the advantage is 0: in particular, $\Pr[Exp(0)=1] = \Pr[Exp(1)=1] = 1$, so $|\Pr[Exp(0)=1] - \Pr[Exp(1)=1]| = 0$.  Since the advantage is 0, this attack strategy is ineffective -- exactly as you'd expect.