$\DeclareMathOperator\Mp{Mp}$Let $F$ be a number field and $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A}_F)$.

Then we can think a submodule $L_{\pi}^2$ of $L_{\mathrm{disc}}^2(\Mp_2)$, the discrete spectrum of automorphic functions on $\Mp_2(F) \backslash \Mp_2(\mathbb{A})$, as

$L_{\pi}^2:=\sum_{a \in F^{\times} \backslash F^{\times^2}} \Theta(\pi \otimes \chi_a)$, where $\chi_a$ is the quadratic character of $\mathbb{A}^{\times}/F^{\times}$ associated to the quadratic extension $F(\sqrt{a})/F$ by global class field theory and $\Theta$ is a theta lift from $\operatorname{SO}_3 \simeq \operatorname{PGL}_2$ to $\Mp_2$.

Then theorem of Waldspurger asserts that $L_{\pi}^2$ is the full near equivalence class in $L_{\mathrm{disc}}^2(\Mp_2)$ such that each irreducible summand is in the global Waldspurger packet $A_{\pi}$ of $\pi$ and the number of irreducible summand of $L_{\pi}^2$ is half the number of $A_{\pi}$.

Since $\pi$ is irreducible cuspidal of $\operatorname{PGL} _2$, the number of $A_{\pi}$ should be finite. Therefore, I think there are only finitely many Hecke quadratic characters $\chi$'s such that $L(\frac{1}{2},\pi \times \chi) \ne 0$ because $\Theta(\pi \otimes \chi) \ne 0$ is equal to $L(\frac{1}{2},\pi \times \chi) \ne 0$.

However, the paper of Friedberg and Hoffstein (Theorem B in https://www.jstor.org/stable/2118638) claims that there are infinitely many quadratic characters $\chi$ such that $L(\frac{1}{2},\pi \times \chi) \ne 0$. So I think it contradicts to $\sharp A_{\pi} <\infty$.

What is wrong in this reasoning?

Any comments are welcome!