Why do primes dislike dividing the sum of all the preceding primes?

Here is a heuristic argument that there is nothing to explain:

The probability that $p$ divides the sum of the preceding primes is $1/p$. So the expected number of primes less than $10^9$ with this property is $\sum_{p \leq 10^9} \frac{1}{p}$. Using Mertens' second theorem, $$\sum_{p \leq 10^9} \frac{1}{p} \approx \log \log 10^9 + M \approx 3.3$$

Here $\log$ is natural log and $M \approx 0.26149$ is Mertens' constant.

This is an example of the motto "$\log \log x$ goes to infinity but has never been observed to do so". It is quite common for people to look at primes $p$ which divide some quantity $a_p$ and conclude that they are surprisingly rare when, in fact, they are simply growing as $\log \log N$ for the reason above.

Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{-1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/articulos.html

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,