Layman here. EE and BS physics. I am "content" in viewing photons/electromagnetic radiation as an "emergent" property of spacetime? due to the electrons ("particles..?") and all their activity jumping around in atoms. [or is this likely too simple?] ... in the "same" manner that sound and water waves are an "emergent" property of air and water particles. OK. So WHAT do neutrinos represent? Can I visualize them as an emergent property of neutrons and protons [and quarks?] and all their activity in the nucleus? Neutrinos are a real mystery to me, as if everything else weren't!
Your second sentence is not correct in our current understanding of physics. Photons are not an "emergent property of space-time" - photons appear even when we take space-time to be a fixed object. Photons are "not due to electrons... jumping around" - they can be created by the motion of charges but they are independent objects. Photons are just as real as electrons and just as fundamental. They do not have any relationship like that of water waves and water molecules.
With that understood, neutrinos are also understood to be independent fundamental particles. If you wish to conceptualize neutrinos, just think of them as almost massless electrons with no electric charge.