Recent data on strongly correlated biological systems show the validity of scaling laws as one of the fundamental traits of collective behaviour. Field theoretical techniques, such as the Renormalization Group, thus became useful tools to describe living systems as bird flocks, cell colonies and insect swarms. Experiments unveiled traces of critical dynamics in the latter system, exhibiting an inertial dynamics in the velocities and a dynamical critical exponent z ~ 1.2. To rationalise this evidence, we develop an inertial active field theory in which the velocity is coupled to its generator of internal rotations, namely the spin, through a mode-coupling interaction. Supported by the indication of weak density fluctuations in insect swarms, we study its near-critical regime with a one-loop Renormalization Group approach under the assumption of incompressibility. The presence of friction in the dynamics of the spin rules a paramount crossover between two fixed points: the unstable underdamped fixed point with z = 1.3 and the stable overdamped fixed point with z = 1.7, where dissipation takes over. We show how finite-size systems with weak dissipation, such as swarms, can actually exhibit the critical dynamics of the unstable fixed point thus providing a theoretical result which is in fair agreement with experimental data.