By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}.$$ These spaces are "formal" over the rationals, meaning that their cohomology ring completely encodes their rational homotopy type.

For $n = 2$ there is no need to involve operads, because there is a direct map $H^*(\operatorname{Conf}_k(\mathbb{C})) \xrightarrow{\sim} \Omega^*(\operatorname{Conf}_k(\mathbb{C}))$, sending the usual generators of the cohomology to $d\log(z_i - z_j)$ (Arnold, 1969). For $n \ge 3$ this doesn't work because the Arnold relations do not hold "on the nose" in $\Omega^*(\operatorname{Conf}_k(\mathbb{R}^n))$.

The proofs I know for $n \ge 3$ all involve operads. More precisely, the little disks operads, whose components are homotopy equivalent to the configuration spaces above. The theorem is that these operads are formal. Hence the configuration spaces are formal. See the results of Kontsevich (and Guillén Santos–Navarro–Pascual–Roig to descend to $\mathbb{Q}$), Tamarkin (for $n = 2$ actually), Lambrechts–Volić, Petersen, Fresse–Willwacher...

I am curious, is there a proof somewhere in the literature that has nothing to do with operads? I have not found any, but, well, I cannot be 100% sure...