There is then the Crown items designated by Queen Victoria. Queen Victoria intended these to be worn by future Queens but I am not clear whether she added any criteria as to whether they are owned by the state. I find that unlikely.
I take your point about the jewels in the Louvre, although arguably one can compare that to the Crown Jewels on permanent display in the Tower of London. Each summer, many major items of the Queen's privately owned jewels are on display at Buckingham Palace in the ballroom. I visited back in 2017 and the Girls of Great Britain tiara and the Cullinan were on display.
My earlier comment about museums typically putting the same small number of high profile items on display to maximise interest is probably still relevant - the average person visiting Buckingham Palace wants to see the tiara the Queen wears on the money or that big diamond she is sometimes spotted wearing. Buckingham Palace organisers may take the view that very few people want to see a tiara that no one has worn for 70 plus years.
As for releasing items about personal jewels, I cannot say why the Palace is reluctant to comment. Lets be fair though they have cooperated with a number of authors over the years but yes there are still mysteries. However I must say that is the Queen's prerogative, she is not compelled to release details about her personal property anymore than you or I.