(Just for notation, I let $\mathrm{Conf}_n(X) = \{ x \in X^n \mid \forall i \neq j,\, x_i \neq x_j \}$. This is what Bezrukavnikov calls $X_n$ and what Kohno–Oda call $F_{0,n}(X)$.)
The issue is in §3, on page 208, when they claim that the monodromy representation $\rho : \pi_1(\mathrm{Conf}_n(\Sigma_g) \to \mathrm{Aut}(\pi_1(\Sigma_g - \{p_1, ..., p_n\})^{ab})$ is trivial. They claim that $\pi_1(\mathrm{Conf}_{n+1}(\Sigma_g)) \to \pi_1(\mathrm{Conf}_n(\Sigma_g))$ has a section. Unfortunately, this is not correct if $g \ge 2$ and $n \ge 2$. (For the torus it exists because it's a Lie group so you can define one explicitly, see "Configuration spaces" by Fadell–Neuwirth.)
Here I should give some credit as I didn't find the issue on my own: see the last remark in "Braids on surfaces and finite type invariants" by Bellingeri and Funar, and a remark in Hain's "Infinitesimal presentations of the Torelli groups".
To see which statement is equivalent to the Koszulity of the universal enveloping algebra, I'll try to rephrase everything in slightly more "modern" language (at the very least, language that I am more familiar with). Bezrukavnikov uses a convention (the same as in the book Quadratic Algebras of Polishchuk–Positselski) that a dg-algebra $A$ is quadratic if it is generated by $A^1$ and the kernel of $T(A^1) \to A$ is generated by elements in degree $2$. If $A$ is graded commutative, then the Koszul dual $A^!$ is the enveloping algebra of $\mathcal{L}$, where $\mathcal{L}$ is the free Lie algebra on the dual $(A^1)^\vee$ modded out by the image of the dual of the product $(A^2)^\vee \to (A^1)^\vee \otimes (A^1)^\vee$. In the case where $A = H^*(X)$, this is exactly the holonomy Lie algebra $\mathfrak{g}_X$ defined by Kohno–Oda.
There are various equivalent definitions of Koszulness of a quadratic DGA $A$. One of them is that the Koszul complex $(A \otimes A^¡, d)$ is acyclic, where $A^¡$ is dual to $A^!$. The Koszul complex of $U(\mathfrak{g}_X)$ is exactly the one called $R(X)$, and Kohno–Oda prove that it is acyclic in the proof of Lemma 4.1, using the section claimed above.
(Maybe a quick remark: nowadays, algebras which aren't $1$-generated can also be considered Koszul. Instead of asking $A$ to be generated by $A^1$ with relations in degree $2$, you can ask $A$ to be generated by some set of generators and ask for relations of weight $2$ with respect to the word length. Then whenever you see an $\operatorname{Ext}$ or something, it is weight graded, and whenever there is a condition of the type "generated in degree $1$" you can replace it by "generated in weight $1$".)