Uniform non-vanishing of prime quadratic twists of standard and symmetric square $L$-functions

Let $S_k$ be the space of cusp forms of weight $k$ for ${\rm SL}_2(\mathbb Z)$, with $\dim S_k\geq1$, and let $χ_p=\left(\frac{\cdot}{p}\right)$ for odd primes $p$. For fixed $s$ with ${\rm Re}(s)>(k+1)/2$, we prove that the proportion of primes $p\leq X$ satisfying $L(h,χ_p,s)=0$ tends to zero as $