The observed distribution of inclinations of long-period comets are why it's called the Oort cloud and not the Oort belt..

Short period comets are relatively close to the ecliptic, while very-long-period comets have evenly distributed inclinations suggesting that the objects in the Oort are more or less spherically distributed.

Discussed here: https://en.wikipedia.org/wiki/Oort_cloud#Development_of_theo...