Regev's LWE Survey contains a sketch of the proof.
Algorithms. One naive way to solve LWE is through a maximum likelihood algorithm. Assume for simplicity that $q$ is polynomial and that the error distribution is normal, as above. Then, it is not difficult to prove that after about $O(n)$ equations, the only assignment to $s$ that "approximately satisfies" the equations is the correct one. This can be shown by a standard argument based on Chernoff’s bound and a union bound over all $s\in\mathbb{Z}_q^n$. This leads to an algorithm that uses only $O(n)$ samples, and runs in time $2^{O(n \log n)}$. As a corollary we obtain that LWE is "well-defined" in the sense that with high probability the solution $s$ is unique, assuming the number of equations is $\Omega(n)$.
It may also be useful to view LWE from a different perspective --- namely as a parameter range for SIS where it is a priori unclear whether a solution should exist or not, so one "plants" one. See these notes for some perspective along these lines.
Note that for standard SIS many solutions exist with high probability, so "LWE = Decisional SIS (in some parameter range)" is easy to get confused by if one is not careful, see for example this question.
