not sure what EH really believe outside of their official line! It seems to me that a short tunnel would only help the current situation of traffic in the immediate Stonehenge area, and wouldn't do much to help the area be put into some better perspective. I've not yet heard them talk about 'spirit of place'.
And of course there are others saying..... hold on, we are finally close to sorting out a World Heritage Site so let's take this once in a lifetime chance to re-route all roads outside of the heritage area. I think this though is unlikely, whereas a longer tunnel is possible. Although it is a lot of money the clout of being one of the most famous World Heritage Sites is not to be underestimated.