Counting Rational Points on the Stacky ${\mathrm{Sym}}^2{\mathbb{P}}^1$

We prove the weak form of the generalized Batyrev-Manin-Malle conjecture formulated in \cite{ellenberg2021heights} for the stack ${\mathrm{Sym}}^2{\mathbb{P}}^1:=({\mathbb{P}}^1\times {\mathbb{P}}^1)\mathrm{/}{S}_2$, where the $S_2$ action just permutes the two coordinates. In particular, we show that the diagonal $\mathrm{\Delta }\subset {\mathrm{Sym}}^2{\mathbb{P}}^1$ is an accumulating substack.