Each individual can have a unique identifier that is not linked to their identity. It could be a public key that they generate from their private key. We'd have to figure out the details of how that would work exactly, i.e. making sure people cannot vote twice, but that's possible to figure out.

I don't think there's any way to do this without giving trust to some central entity without a public blockchain. The public blockchain does become the source of trust, but you need a resource (like energy), that is finite, to secure that network.

However, going back to your original comment, you said we are assuming a zero carbon footprint, so that would assume some future where this trust could be put on a network in a way that would be environmentally sustainable. All I'm saying is that there are more possibilities to this technology than I think you are acknowledging.