How do we prove the time complexity of this simple problem in probabilistic inference on a Bayesian network?
Exponential time is not required for finding the requested probability.  In fact, only linear time is needed, as we will see below.

First let's define some additional notation.  Let $X = (x_1, x_2, \ldots, x_n)$ and $Y = (y_1, y_2, \ldots, y_n)$ -- that is, two $n$-tuples of our variables.  Let $\mathbf{1}$ denote an $n$-tuple of 1s, $ (1, 1, \ldots, 1)$.  And let $S_n$ denote the set of all possible $n$-tuples of $0$ and $1$.  In CS literature, the elements of $S_n$ are sometimes called "strings".  If $\sigma$ is one such element, we write $\sigma(k)$ for its $k$th component.

In this new notation, the task is to compute $\Pr(Y = \mathbf{1})$.  We can marginalize out the configuration of 0s and 1s on $X$ as follows:
$$\Pr(Y = \mathbf{1}) = \sum_{\sigma \in S_n} \Pr(Y = \mathbf{1} | X = \sigma) \Pr (X = \sigma).$$

From the prompt we know that each $x_k$ is 0 or 1 with equal probability, so $\Pr (X = \sigma) = 2^{-n}$ for every $\sigma \in S_n$.  Additionally, for any given $\sigma \in S_n$, we know that $\Pr(Y = \mathbf{1} | X = \sigma) = \prod_{k=1}^n p_{k,\sigma(k)}$, since the probabilities $p_{k,\sigma(k)}$ are independent.  Combining these observations, we obtain
$$\Pr(Y = \mathbf{1}) = 2^{-n} \sum_{\sigma \in S_n}  \prod_{k=1}^n p_{k,\sigma(k)}.$$

Our next step is to show that we can factor the left-hand side of this equation as follows:
$$\sum_{\sigma \in S_n}  \prod_{k=1}^n p_{k,\sigma(k)} = \prod_{k=1}^n (p_{k,0}+p_{k,1}).$$
We prove this inductively.  When $n = 1$, both sides are $p_{1,0}+p_{1,1}$, so the base case holds.  For the inductive case, we assume the $(n-1)$th case and write the $n$th case as 
$$(p_{n,0}+p_{n,1}) \sum_{\sigma \in S_{n-1}}  \prod_{k=1}^{n-1} p_{k,\sigma(k)} = (p_{n,0}+p_{n,1}) \prod_{k=1}^{n-1} (p_{k,0}+p_{k,1}).$$
Both sides simplify to those of the above identity, respectively, so the identity is proved.

This leaves us with the result that $\Pr(Y = \mathbf{1}) = 2^{-n} \prod_{k=1}^n (p_{k,0}+p_{k,1}).$  For a hash table representation, look-up times are $O(1)$ for each $p_{k,0}$ and $p_{k,1}$, so we can compute this product with a simple for-loop in a runtime of $O(n)$.