In reply to mondite:
Ok, I'm going to stick my head above the parapet and offer the flipside, and comment on the matter of access on the bridge, not the wider issue of public benefit as a result of IHT and charitable status.
Map: https://maps.northyorks.gov.uk/connect/analyst/mobile/#/main?mapcfg=Out_and...
The bridge is a permissive path, the clue is in the description of the path. That makes it different from a normal right of way. Without knowing the terms of the agreement of the permissive path, it's difficult to comment.
I am local (ish) to Bolton Abbey. I happen to be a charity trustee of a charity a long way from Yorkshire which is a sizeable estate, a lot of it arable & farmland (including conservation and land management consideratons) with a lot of rights of way and permissive paths. The trustees do our utmost to encourage public access and engagement, but we have to juggle many balls at different times. I'd bet there are quite a few other charity trustees on UKHW who also have to balance priorities for their charities that might lead to a temporary outcome that they prefer was otherwise.
Post edited at 10:02