Why does DM not being able to radiate energy keep it from total collapse?

When ordinary matter (like interstellar gas) begins to collapse via gravity, the gas atoms collide with one another and thermalize or exchange kinetic energy in those collisions and the entire condensing cloud strives towards thermal equilibrium. As gravity pulls the gas in closer, its density climbs, the collisions become more frequent and more energetic, and the gravitational potential energy gets dissipated as heat- and so the temperature of the cloud goes up. That temperature rise causes the pressure of the gas to rise and that pressure resists further collapse.

Eventually the temperature of the cloud rises enough that it begins radiating like a black body, allowing the thermal pressure to go down. The dissipation of the compressive heating allows gravity to pull the gas in closer, and as long as the gas can radiate its heat away, the collapse can proceed and the cloud gets smaller and denser.

The key to getting this process to go is the gas particles' ability to interact with other particles and dissipate heat, which is electromagnetically mediated via photon exchange. And here is the reason this process (thermalization & dissipative collapse) doesn't work for dark matter particles: because they do not experience the electromagnetic force, they cannot thermalize nor can they radiate and that means they are resistant to gravitational collapse.

Stars collapse into black holes precisely because they have means of radiating away their internal energy. This removes internal pressure support resulting in contraction/collapse. .

In the case of core collapse supernovae the internal energy is first used to dissociate heavy nuclei (electromagnetic and strong force interactions) and then by neutrino emission as free neutrons are produced (weak interactions).

The neutrinos are thermal, so each one carries away $sim kT$ of energy. Since basically all the core baryons turn into neutrons with the emission of a neutrino, then almost all the thermal energy escapes on the neutrino diffusion timescale, which is a matter of seconds.

Supermassive black holes grow in a different way, by accreting material that gets close to their event horizons. Yet there is a similar requirement for that material to shed angular momentum and this can only be done through viscosity of some kind, mediated by electromagnetic interactions.

Dark matter does not have these routes to losing its kinetic energy or angular momentum and is therefore unable to form collapsed configurations.