Attractive interaction in BCS theory

The common way to discuss BCS superconductivity, as I have seen in several books, is to assume that we have a Fermi liquid, on top of which we add an interaction of the form
$$H_{int}=sum_{p,p^prime,q} V(q)psi^dagger_uparrow(-p) psi^dagger_downarrow(-p^prime) psi_downarrow(p^prime+q)psi_uparrow(p-q)$$
The assumption is that as long as $V$ is negative we get a Cooper instability and a condensate will form at low enough temperatures. This doesn’t make sense to me as if $Vpropto 1/q^2$ (in 3D) we just get a correction to the Coulomb interaction and remain with a Fermi liquid.

In that case, what do we actually need to demand on $V(q)$ to get superconductivity? Do we explicitly need a constant term?