The problem of the theoretical description of the critical temperature Tc of a Fermi superfluid dates back to the work of Gor’kov and Melik-Barkhudarov (GMB), who addressed it for a weakly-coupled (dilute) superfluid in what would today be referred to as the (extreme) BCS (weak-coupling) limit of the BCS-BEC crossover. The point made in this context by GMB was that particle-particle (pairing) excitations, which are responsible for superfluidity to occur below Tc, and particle-hole excitations, which give rise to screening also in a normal system, get e ectively disentangled from each other in the BCS limit, thus yielding a reduction by a factor 2:2 of the value of Tc obtained when neglecting screening e ects. Subsequent work on this topic, that was aimed at extending the original GMB argument away from the BCS limit, has tout court kept this disentangling between pairing and screening throughout the BCS-BEC crossover, without realising that the conditions for it to be valid are soon violated away from the BCS limit. Here, we reconsider this problem from a more general perspective and argue that pairing and screening are intrinsically entangled with each other along the whole BCS-BEC crossover but for the BCS limit considered by GMB, with the particle-hole excitations soon transmuting into particle-particle excitations away from this limit. We substantiate our argument by performing a detailed numerical calculation of the GMB diagrammatic contribution suitably extended to the whole BCS-BEC crossover, where the full wave-vector and frequency dependence occurring in the repeated two-particle in medium scattering is duly taken into account. Our numerical calculations are tested against analytic results available in both the BCS and BEC limits, and the contribution of the GMB approach to the scattering length of composite bosons in the BEC limit is highlighted. We calculate Tc throughout the BCS-BEC crossover and find that it agrees quite well with available Monte Carlo calculations in the unitarity regime.