There's a cattle grid on it, so I'd imagine that most horse routers will try and avoid that :)
If it was a locked gate on a private bit of road and the bridleway actually ran to the side, then a bunch of "no" tags would make sense, but presumably foot access isn't actually illegal across the cattle grid, and people are free to hop across on foot (or use the gate) as required?
I'm not overly familiar with the area and can certainly think of places in Yorkshire where a private cattle crid has been added in parallel with a gate on a public road (with access rules posted about the cattle grid).