The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the Emden–Fowler nonlinear differential equation, $$f''(t)=t^{p}[f(t)]^q,$$ for $p=1$, $q=-1$. A particular solution is $$f(t)=\lambda t^{(p+2)/(1-q)},\;\;\lambda=\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$ One readily checks that the asymptotic limit $x_n\rightarrow \sqrt{3n}$ for $x_n=f'(n)$ is obtained for $p=1$, $q=-1$.
This connection to the Emden-Fowler equation motivates the following
Conjecture: The recursion relation
$$x_{n+1}=x_n+n^{p}(x_1+x_2+\cdots+x_n)^q$$
has for $p\geq 0$, $q\leq 0$, $p+q>-1$, the limit
$$\lim_{n\rightarrow\infty}n^{(1-q)/(1+p+q)}x_n=\frac{p+2}{1-q}\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$
For $q=-1$ this is stated as an open problem on page 11 of Sequences of Real Numbers, by Sîntămărian & Furdui.
Here are two numerical tests (blue is the recursion for $x_n$, gold the conjectured large-$n$ asymptotic):
