They charge to go into Cathedrals nowadays.
I personally refuse to pay to enter any CoE Cathedral on the grounds that "I am a life member" and have told various clergy at entrance points this over the years.
I would be happy to pay if I wanted to visit a house of worship where I was not a member (RC Cathedral or a Synagogue for example) as I would be going in as a tourist.
Having to pay to go into a museum or art gallery on the other hand is a different kettle of fish. We have already paid for these places and the exhibits therein. Privately owned collections I am happy to pay for as I pay to visit museums and art collections when I am abroad but I would object to being charged to view items that this nation has paid for over the years. All such places have been handsomely endowed by their founders but I just don't see picking up the tab for years of mismanagement of the original funds.
Sorry Boris but on this one we must agree to differ.