In terms of access I believe paper roads function pretty much equivalently to public roads. This makes a bit more sense if we use their actual term - paper road is just the colloquial. They are just unformed legal roads.
https://www.herengaanuku.govt.nz/types-of-access/unformed-legal-roads
What I don't know is whether all roads are obliged to allow vehicles of all types or not; I would assume not as there are plenty that are marked not suitable for either weight, length or both. And it would be interesting to know if the vehicle has to be in road legal condition - ie normal tyres at road inflation etc. Beyond that though most of the unformed roads I know of are wholly unsuited for vehicles other than quad bikes, tractors etc. Their gradient is often well beyond standard, and they're not even gravelled like a farm track let alone formed with a nice hump to keep water draining off into the non-existent ditches on either side.
Usually they're just some lines on a map relatively devoid of the context of topography.
Despite the interest the bylaw received - thanks to a very well made campaign against it - this most likely impacts a fraction of the 3500* people that submitted, and of those its really only a fraction that have been tearing up the land either side of the road. So the solution of just blocking it to vehicles seems to me to be the most pragmatic.