|Stonehenge when I first visited and you weren't kept out|
First of all, it's silly to think that somehow you would be able to lose yourself in neolithic times if only the road were hidden. That might have been the case in my youth, when you could wander around the stones to your hearts content. But now you are kept a good distance away by a fence (unless you have some ridiculous druidical religious belief (largely made up in a pub in London in Victorian times - I've seen the plaque), in which case you can trample all over the ancient monument that has nothing to do with your 'religion'). This forces a visit to Stonehenge to be a 21st century experience, so just accept it.
The reason I say the idea of burying the road gets things back to front is that I think it takes the wrong viewpoint. I drove down the A303 to and from the Brympton Festival on Sunday and it was totally wonderful getting that heart-stopping view of the monument as I drove past each way. If they put the road in a cutting or tunnel, thousands of people a day would miss out on that stunning view of Stonehenge - and I think that outweighs any prissy attempt to make the visitor experience better.
Keep that wondrous view from the road, please!