You asked for a heuristic answer.

There is an heuristic argument that infinitely many such partial sums should exist.  Consider $P(k)$, an heuristic estimate of the probability that the partial sum of the first $k+1$ primes would be divisible by $p_k$.  Now $$p_k \sim \frac1{k \log k}$$ and if only random chance were involved,  $$P(k) \approx \frac1{p_k}  \sim \frac1{k \log k}$$ 

In that case, the expected number of primes with the property you want would be something like
$$\int_2^\infty \frac1{x \log x}\,dx$$
and that integral diverges to infinity.

The reason it seems so rare is that the rate of divergence is like $\log(\log x)$ and while that function goes to infinity, "nobody ever sees it do so."

On the other hand, **proving** that there an infinite number of such values of $k$ (in the same sense that Euclid's argument proves there is no last prime) is probably quite difficult.  And if the conjecture that there are only a finite number of such values of $k$ turned out to be false,  **proving** that some particular $k$ is the last one with this property would seem to be even harder.