Yeah OG voxels are nice and I don't know why more people use them, but it's probably because it's somehow harder to do them with current things. I might be wrong but I think the OG ones aren't even "billboarded" but literally just 2D rectangles and then scaled based on perspective. Really wish there was a different term for "flat" voxels and the "full 3d" ones.
In unity however you can't really tell shaders "draw a rectangle" and everything kinda has to go through a mesh. What I do is I have a script that first take a voxel object (.vox from MagicaVoxel) and from each voxels I make a quad, so the whole object becomes a mesh made of individual floating quads. The colors get saved into vertex colors and local positions into an UV. Then the shader does the billboarding (by doing some arcane matrix operations with the local positions that I may or may not have hacked together from some forum posts) and coloring.
I think a more "true to the original" way to do this would be with GL.QUADS but I'm not sure if it worth it.
At some point I'd like to share it all as a package but it's kind of a finicky and arcane process right now and there are a bunch of unused / experimental things floating in the code that might be confusing, but that's the gist of it.