BIG UPDATE.

I decided to just email the Elveden Estate office about this, and they confirmed that the closure is just for motor vehicles :) I've uploaded screenshots of the whole exchange below:

Part 1: https://i.imgur.com/YnbvbC4.jpeg
Part 2: https://i.imgur.com/qYWlIYd.jpeg
Part 3: https://i.imgur.com/EqZ9itd.jpeg
Part 4: https://i.imgur.com/hXFgGrj.jpeg

Hopefully you can still get the council to clarify the wording on their website, but as far as OSM is concerned I think this settles this issue :)