I think you misunderstand the difference between Queen and state. The Queen is independently very wealthy and is responsible for the upkeep of her own personal palaces. However Buckingham Palace is owned by the State and is effectively the Queen's office and place where she carries out official functions in London hence it is maintained by the State. Of course the Queen also entertains at her personal house at Windsor Castle on behalf of the nation too!
Also, the money the Royal family from the civil list historically doesn't come from taxpayers but is actually from the revenues of the Duchy of Lancaster which were always privately held by the monarch since the time of Henry IV who inherited them from his mother, Blanche of Lancaster.
Bookmarks