  We extend the recently-introduced weak Bruhat interval modules of the type A
$0$-Hecke algebra to all finite Coxeter types. We determine, in a
type-independent manner, structural properties for certain general families of
these modules, with a primary focus on projective covers and injective hulls.
We apply this approach to recover a number of results on type A $0$-Hecke
modules in a uniform way, and obtain some additional results on
recently-introduced families of type A $0$-Hecke modules.
