What theory supported that the collapse of a massive star's iron core produces neutrinos?
A massive iron core collapse requires converting protons in neutron which results in neutrino emission.
A massive star's iron core needs to resist collapse under gravity. When the core undergoes fusion reactions, this resists gravitational collapse. Once fusion stops, the core collapse is stopped by electron degeneracy pressure. This is effectively the Pauli exclusion principle which forbids two electrons being in the same quantum state.
If the core has a mass of over about 1.4 solar masses electron degeneracy pressure can no longer stop gravitational collapse. The core at this stage collapses into a neutron star.
In order for the neutron star to form electrons and protons combine to become neutrons. In order to preserve baryon numbers the a neutrino is emitted in the process.
Hence the formation of a neutron star produces vast numbers of neutrinos.