Normality of $(1-\sum_{a\in A}2^{-a})^{-1}$ for infinite primitive subsets $A\subseteq\mathbb N$

Let $\mathcal P$ denote the set of prime numbers, and consider $$ N =\frac{1}{1-\sum_{p\in\mathcal P}2^{-p}}. $$

Numerically, the binary expansion of $N$ appears to behave like that of a base-$2$ normal number. For example, among the first $10^6$ binary digits, the frequencies of $0$ and $1$, and more generally of short binary blocks, are close to the frequencies predicted by normality.

I observed similar behavior after replacing $\mathcal P$ by several other primitive sets $A\subseteq\mathbb N$, where “primitive” means that no member of $A$ divides another, and considering $$ \frac{1}{1-\sum_{a\in A}2^{-a}}. $$

The behavior becomes visibly less random for extremely sparse sets.

For example, define the sequence $(a_m)_{m\geq 0}$ by $$ a_0=1,\qquad a_{m+1}=2^{a_m}, $$ and let $$ A={p_{a_m}:m\geq 0}. $$

Thus the indices of the primes in $A$ are $$ 1,2,4,16,65536,2^{65536},\ldots. $$

For this set, the initial binary digits show much more visible structure. This may be related to the enormous gaps between successive elements.

Is there a heuristic reason to expect $x_A$ to be normal in base $2$ for sufficiently dense or irregular primitive sets $A$? Are there any known results relating the additive, density, or lacunarity properties of $A$ to the normality, or even simple normality, of $$ \frac{1}{1-\sum_{a\in A}2^{-a}}? $$