In the infinite case, there is a close notion of "locally indicable group", i.e. a group where every finitely generated subgroup maps onto $\mathbb Z$ (see, for example, this paper). Locally indicable groups are left (right) orderable, hence important. Note that in that notion, not all subgroups are considered but only finitely generated, and "non-perfect" is replaced by a stronger property "maps onto $\mathbb Z$". But in the finite case all subgroups are finitely generated, and "maps onto $\mathbb Z$" is an infinite analog of "maps onto a finite cyclic group" (= non-perfect). So "locally indicable" is possibly the infinite analog of the property you consider.
In the infinite case, there is a close notion of "locally indicable group", i.e. a group where every finitely generated subgroup maps onto $\mathbb Z$ (see, for example, this paper). Locally indicable groups are left (right) orderable, hence important. Note that in that notion, not all subgroups are considered but only finitely generated, and "non-perfect" is replaced by a stronger property "maps onto $\mathbb Z$". But in the finite case all subgroups are finitely generated, and "maps onto $\mathbb Z$" is an infinite analog of "maps onto a finite cyclic group" (= non-perfect). So "locally indicable" is possibly the infinite analog of the property you consider.