Personally I get the feeling they're holding back because there's a strong indication it's modern. As you said, its not really that big a deal to get it tested and it would answer many questions.
To me it looks all the world like an amateur attempt at making a Medieval arrow. Something you'd see at a reenactment event for instance. A couple of experts have commented on how incredibly well made the head is (and how much it looks like Cole's WA/type 16) and yet the binding looks terrible. Thick linen unevenly bound with the fletchings far too close to the nock smacks of somebody's first try.
If it was genuine, in my opinion there's a huge juxtaposition - with such a badly made fletching and nock end, the head doesn't match. You wouldn't put a beautifully made, time consuming and highly expensive barbed head on such a crappy arrow in times when it mattered. If you're a reenactor however, making crappy arrows and buying the best heads on the internet is very common, as is using linen for the binding.