I don't know any answers here, but this is an awesome question. I too am now super-curious about the Cosmic Neutrino Background.

https://en.wikipedia.org/wiki/Cosmic_neutrino_background

...And I suppose there are probably good reasons for this to be impossible, but wouldn't it be wild if a "mechanism" for things like the "randomness" of beta decay were that a really slow/low energy neutrino from the big bang interacts with a neutron, causing it to decay into a proton, and an electron, and the neutrino gets a boost in energy as well.