EDIT 3 : Sorry for editing this old answer one more time, but I want to also point out for future readers that there is a proof that Dedekind domains have stable range $2$ which is very similar to my proof for PID's in Satz K.13 of the book *Algebra* by Jantzen and Schwermer. The whole Appendix K of that book is a lovely introduction to the whole notion of stable range.

EDIT 2 : I just learned of a super-short proof of the special case of the Bass Stable Range theorem alluded to in the question (the one giving the stable range for Noetherian $d$-dimensional rings). It's a little more abstract than what I did below for PID's, but not much harder. See Section 2 of

MR0217052 (36 #147)
Estes, Dennis; Ohm, Jack
Stable range in commutative rings.
J. Algebra 7 1967 343–362.

EDIT : Here's a proof that works for $R$ a a PID, which implies that the condition of generating the unit ideal is the same as having gcd equal to $1$.

For some $n \geq 2$ consider a tuple $(a_1,\ldots,a_{n+1})$ of elements of $R$ whose gcd is $1$. We want to find $r_1,\ldots,r_n \in R$ such that $\text{gcd}(a_1+r_1 a_{n+1},\ldots,a_n + r_n a_{n+1}) = 1$.

There are three cases. If $a_{n+1}=0$, then there is nothing to do. If $a_i=0$ for some $1 \leq i \leq n$, then we can take $r_i=1$ and $r_j=0$ for $j \neq i$.

The most interesting case is when none of the $a_i$ equal $0$. In this case, we will only need $r_1$ (the rest of the $r_i$ can be taken to be $0$). Set $b = \text{gcd}(a_2,\ldots,a_n)$,
and let $p_1,\ldots,p_k$ be the distinct primes dividing $b$. For each $i$, we know that $p_i$ cannot divide both $a_1$ and $a_{n+1}$. This implies that there exists some $c_i \in \{0,1\}$ such that
$$a_1 + c_i a_{n+1} \neq 0 \quad (\text{mod } p_i).$$
By the Chinese remainder theorem, there exists some $r_1 \in R$ such that
$$r_1 = c_i \quad (\text{mod } p_i)$$
for $1 \leq i \leq k$, which implies that
$$a_1 + r_1 a_{n+1} \neq 0 \quad (\text{mod } p_i)$$
for all $1 \leq i \leq k$. We conclude that the gcd of $a_1+r_1 a_{n+1}$ and $b$ equals $1$, and thus that the gcd of $a_1+r_1 a_{n+1},a_2,\ldots,a_n$ is $1$.

Here is what was my original answer:

This does not exactly answer your question, but it is much easier to prove that the complexes that van der Kallen needs are highly connected for $\mathbb{Z}$ than for general rings. This was originally done by Maazen in his unpublished thesis, which can be downloaded here. There is also a different proof of this connectivity in Step 2 of the proof of Theorem B in my paper "The complex of partial bases for $F_n$ and finite generation of the Torelli subgroup of $\text{Aut}(F_n)$" with Matt Day, available on my webpage.