I would like to contribute in outlining two natural generalizations of the original question for which the Jacobson radical also comes into play. Eventually, I would like to quote two research results for which OP’s question is essential.

Considering that $R^{\times} = GL_1(R)$, it is natural to ask **"When does the surjection $R \twoheadrightarrow R/I$ induce a surjective map $GL_n(R) \rightarrow GL_n(R/I)$?"**. Here $I$ is an ideal of $R$ and the second map is the reduction of matrix coefficients modulo $I$. If $I$ is an ideal contained in the Jacobson radical $\text{rad}(R)$ of $R$, then $GL_n(R) \rightarrow GL_n(R/I)$ is surjective [4, Exercise I.1.12.iv].

Considering that $R$ is an $R$-module and that every element in $R^{\times}$ generates $R$, it is natural to ask **"For $M$ an $R$-module, when do $R$-generating sets of $M/IM$ lift to $R$-generating sets of $M$?"**. This holds if $I \subset \text{rad}(R)$ and $M/IM$ is finitely generated over $R$ by Nakayama’s lemma (see also the related concept of superfluous submodule).

I introduce now a result published in 2003 which directly relates to OP's question. Let $R$ be a Dedekind domain which is universal for $GE_2$. The latter property means that in the subgroup $GE_2(R) \subset GL_2(R)$ generated by the transvections and the diagonal matrices, the latter generators are only subject to the « obvious » or universal relations (equivalently $K_2(2, R)$ is generated by symbols as a normal subgroup of $St_2(2, R)$ [1, 3]).
Then we have:

If $p$ is a prime element of $R$ such that the natural $R
\twoheadrightarrow R/(p)$ induces a surjective homomorphism between unit groups, then the localization $RS^{-1}$ of $R$ with $S = \{1, p, p^2, \dots\}$ is also universal for $GE_2$.

This was proved in [2] and the result was shown to be sharp in [3].

I can’t resist mentioning a humble result of mine, which I would like to believe is entertaining. A finitely generated group $G$ of rank $n$ is said to satisfy the *generalized Andrews-Curtis conjecture* if every $n$-tuple of elements which normally generate $G$ can be transitioned to an $n$-tuple of generators of $G$ by means of finitely many $AC$-moves, i.e., applying transvections or replacing one component by a conjugate a finite number of times. (If $G$ is the free group on $n$ generators, this is just the Andrews-Curtis conjecture, still unsettled.) Then the result reads as:

The solvable Baumslag-Solitar group $\langle a, b \,\vert\, aba^{-1} = b^n \rangle$ with $n \ge 1$ satisfies the generalized Andrews-Curtis conjecture if and only if the natural map $R \twoheadrightarrow R/(n - 1)R$ where $R = \mathbf{Z}[1/n]$ induces a surjective homomorphism between unit groups.

[1] "On the structure of the $GL_2$ of a ring", P. M. Cohn, 1966.

[2] "Tits systems with affine Weyl groups in Chevalley groups over Dedekind domains", E. Abe and J. Morita, 1988.

[3] "Subrings in quadratic fields which are not $GE_2$", H. Yu and S. Chen, 2003.

[4] "The K-book", C. Weibel, 2013.