Let $f_a : S \to R$ is a family of functions indexed by $a\in P$.
Consider the assumption that $(a, f_a(x))$ is indistinguishable from uniform, over the distribution of $a\leftarrow U$ (uniform) and $x\leftarrow D$ (some efficiently sampleable distribution).
Is this assumption equivalent to, for all but a negligible fraction of $a$, $f_a(x)$ is indistinguishable from uniform over the distribution of $x$?
I'm inclined to think they are. But I'm not very sure and would like a proof.
EDIT: to make this more clear. Let $A$ be a random variable with uniform distribution $U$ over $P$, and $X$ be an independent random variable with some efficiently sampleable distribution $D$ over $S$. Also, let $Y$ be an independent uniformly random variable over the codomain $R$.
The 1st assumption says for any polynomial-time distinguisher $M$, consider the random variable $M(A, f_A(X))$ and $M(A, Y)$, then $$ |Pr[M(A, f_A(X))=1] - Pr[M(A, Y)=1]| \le negl. $$
The 2nd assumption says, there exists a subset $Q \subseteq P$ with $1-|Q|/|P|$ negligible, such that for any $a \in Q$, for any polynomial-time distinguisher $N$, $$ |Pr[N(f_a(X)) = 1] - Pr[N(Y)=1]| \le negl. $$