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access:private added additional to leisure:park may be acceptable, although I don’t know much on mapping places like London. OSM currently has little, if any way, of really splitting up the public/private/…other things that get mapped, therefore some people seem to add things as x:y (public default) then correct it with access=. This ‘add it wrong and correct it’ approach seems strange though, especially considering the frequency of private features.

I have similar opinions to your example of Buckingham palace being a leisure park with tracks and drives being mapped as public roads, and a path around someone’s back garden or pond being marked as a highway:footway. I think all in all, the ‘leisure:park’ tag you mention has been used to get a render, rather than it being correct, so I would encourage you to make osm’s data correct, rather than make it just to render.