That sounds completely wrong. An unadopted private road is still one to which the public would expect access (same as for supermarket car parks) so is therefore subject to the normal traffic rules.
There'd be no reason for the public to access it as it's a dead end and it's just a dirt track off an old cobbled road. The only reason to drive up it would be to park outside a house but there's plenty of room at the end so people don't tend to. The DVLA accepted it so it must be ok.
https://maps.google.co.uk/maps?hl=e...=jXLZXz0dAO06EIRM_07Fhg&cbp=12,82.84,,0,19.87