When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in degree $n-1$, which has a pure Hodge structure of weight $n$. Using a "purity implies formality" principle one can then deduce that $\mathrm{Conf}_k(\mathbf{R}^n)$ is formal. Part of this is certainly folklore since a long time, but to my knowledge the earliest explicit statement of a sufficiently general "purity implies formality" statement to deduce this is in a paper of Cirici-Horel from last year. See Proposition 8.2 of their paper for an explicit statement which includes as a special case formality of these configuration spaces.

But yes, I do believe that (remarkably enough) the first proof of formality of $\mathrm{Conf}_k(\mathbf R^n)$ for $n > 2$ is through the results of Kontsevich and Lambrechts-Volic.

A remark is that the phrase "formality of the little disks" could mean either "the dg operad of chains on $\mathcal D_n$ is formal (as a dg operad)" or "the dg Hopf co-operad of cochains on $\mathcal D_n$ is formal (as a dg Hopf co-operad)". The former statement does not at all imply formality of the individual spaces $\mathrm{Conf}_k(\mathbf R^n)$ since the formality is not required to have any compatibility with cup product. Tamarkin's proof, for instance, only gave formality in the weaker sense, and the proof in Lambrechts-Volic only gives Hopf co-operad formality when $n \geq 3$.

When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in degree $n-1$, which has a pure Hodge structure of weight $n$. Using a "purity implies formality" principle one can then deduce that $\mathrm{Conf}_k(\mathbf{R}^n)$ is formal. Part of this is certainly folklore since a long time, but to my knowledge the earliest explicit statement of a sufficiently general "purity implies formality" statement to deduce this is in a paper of Cirici-Horel from last year. See Proposition 8.2 of their paper for an explicit statement which includes as a special case formality of these configuration spaces.

But yes, I do believe that (remarkably enough) the first proof of formality of $\mathrm{Conf}_k(\mathbf R^n)$ for $n > 2$ is through the results of Kontsevich and Lambrechts-Volic.

A remark is that the phrase "formality of the little disks" could mean either "the dg operad of chains on $\mathcal D_n$ is formal (as a dg operad)" or "the dg Hopf co-operad of cochains on $\mathcal D_n$ is formal (as a dg Hopf co-operad)". The former statement does not at all imply formality of the individual spaces $\mathrm{Conf}_k(\mathbf R^n)$ since the formality is not required to have any compatibility with cup product. Tamarkin's proof, for instance, only gave formality in the weaker sense, and the proof in Lambrechts-Volic only gives Hopf co-operad formality when $n \geq 3$.