IIRC, there is another quantum number, the chirality. All the neutrinos measured are left chiral. The sterile neutrino, which does not pair with other lepton, is right chiral.
So either they aren't produced by beta decay, or they are but they can't be captured by gallium, or we're seeing a sterile neutrino in a different sense. ("Sterile" because it can't interact with gallium, but not sterile due to chirality.)
If I understand, the conversion of gallium into germanium is exactly a beta interaction. (But I may not understand, because it seems to me that gallium->germanium should emit a neutrino, not absorb one.) So I have a hard time seeing how a neutrino could be emitted by the source, but not absorbed by gallium. I could more readily see it as saying that there's a fourth type of neutrino that they cycle through, and while in that fourth state it can't convert gallium. But that wouldn't be "sterile" in the chirality sense.
In a "standard" beta decay the neutron turns into a proton and emits an electron and an anti-electron neutrino. In this reaction its absorbing an electron neutrino and emitting an electron (while still turning the neutron into a proton). Absorbing an electron neutrino is pretty much equivalent to emitting an anti-electron neutrino.
Neutron-Neutrino interaction?
https://www.vivaxsolutions.com/physics/feynman-diagrams.aspx
I even seem to remember that the most commonly produced neutrinos are antineutrinos (so the anti aspect is more fundamental than a convention for each particle type).