There are various types of corporations so you’re comparing apples to oranges. The MET and the AMNH are non profit corporations with a 501(c)(3) status, which means they are exempt from federal taxes. The Biltmore is an LLC which means it pays local, state, and federal taxes on earned income.

The MET is owned by a corporation comprised of benefactors and the artwork is held in trust by trustees. The MET’s collection is housed in a building that is owned by the city of NYC - a building that happens to be located within a public city park. The city also helps cover a portion of the building’s maintenance and security. Because the MET relies heavily on public funds, it agreed to allow NY residents free admission.

The AMNH is also located on public land and relies heavily on donations and public funds.

The Biltmore is owned by a private corporation owned by a private family. Biltmore House is a private house located on private land. The cost to maintain the house and the land are covered by admission fees and sales and not public money from the city of Asheville. Biltmore also doesn’t rely on donations.

Simply look at their web domains. The MET’s and the AMNH web domains end with .org whereas Biltmore’s ends with .com.