Basically the answer is that the Duchy pretends that it is simply a private person (Charles Windsor) and so it shouldn't be subject to any more prying into its affairs than would be the case for you or me. This is sufficiently arguable that it doesn't fear ridicule.
https://en.wikipedia.org/wiki/Littlebredy
It was sold in 1797 to the current family, who was a wealthy banker, so they've always been like this? :D
However, firstly (least importantly) not all land in England is registered. About 10-15% isn't registered. Registration is compulsory for new transfers today, but there are a large tracts of land whose owner hasn't changed for decades or even centuries, registration offers some benefits, but they're not obliged to register unless they want to transfer the property.
Secondly, the paperwork may just say a place is owned by a corporation in a tax haven. Knowing the land my building is on is owned by "Property Holding Corp Sierra Sixty Eighteen" in the British Virgin Islands is almost exactly the same as not knowing who owns it.
(highest value, not largest area)
I make the distinction because as an American, many of my fellows conflate the two.