Yes, some people really need big vehicles (and/or we need them to keep farming) and so you need exemptions. In the UK there is Red
diesel for example that is for farm use / private use that is much cheaper than diesel for the road.
Right—it's not a subsidy, at least in the US, because much or all of the gas tax goes to road maintenance, so if you're not driving on public roads, you don't have to pay it.
Doesn't make sense to make e.g. farm tractors that rarely see a public road, pay for public road maintenance when fuel's purchased for them. Or trucks and other vehicles (some farms are really big and may include road-worthy vehicles that hardly drive on a public road more often that the tractors do).
If you're going to only drive on private land, you also don't need tags/plates for your car, at least around here. No registration. Just personal property tax and you're good to go.