I know it is very hard to decide, given a metric cell complex, if it is nonpositively curved or not. And by the Notes of Sisto, there is a theorem saying: A compact polygonal complex is CAT($0$) if and only if the length of any loop in any link is at least $2\pi$ (Theorem 11.6.2.). Also, a cube complex is CAT($0$) if and only if it is simply connected and all links are flag complexes.
I don't know if links are flag complexes are easier to check than the length of any loop in any link is at least $2\pi$? Is that the reason why we focus on the cube complex rather than other polygonal complexes?
The reason checking that flag complexes is sufficient for a cube complex is because, when the cubes are given the standard unit Euclidean cube metric, the flag complex condition implies the "all link lengths $\ge 2\pi$" condition.
The reason for focussing on cube complexes is more because of the many situations in which they turn up naturally (see the work of e.g. Sageev), and the good algebraic/combinatorial properties that they possess which general $\text{CAT(0)$ spaces do not possess (see the work of e.g. Haglund and Wise).