I have pretty limited ability to comment given I've only been to the Rapidan once. But in that one trip (a guided trip where I rode with the guide in his pickup), I had a few observations that I believe are fairly applicable. I wouldn't dare bring a non-4WD or high ground clearance vehicle up the road we used, although I saw one at the top. I saw no fewer than four pickup truck drivers choose not to use a bridge to cross the stream, and instead chose drive off the road and speed
through river (instead of at least slowly fording it). This was in the brief periods of time that I was actually even near a crossing. Additionally, I saw at least three groups of ATV and/or dirt bike riders that were constantly leaving the trails and trampling vegetation, spraying dirt into the river as their wheels spun, as well as driving across, along, and in the stream. It was unfortunate seeing the blatant disregard for our natural resources. These are all environmentally destructive activities that would be close to impossible, if not completely impossible, with a "normal" car.
The NPS Superintendent cited his responsibility to protect natural resources as one of his reasons for rejecting the request. At the end of the day, he's right - making an area less difficult to access will, in theory, increase the environmental damage due to human interaction. But if he really is concerned about protecting this natural resource, perhaps he should also consider either improving enforcement of existing rules (if they exist) for those that "can" access that area, or re-assessing what rules should be in place to minimize the inappropriate behavior I noted above.