I'd be very interested to hear about any respectable museum that sends off items that are not displayed to be stored into some rich person's attic (or for display at a private residence as I assume you are implying).
If you have a valuable artifact that you don't have on display, renting it for a large sum to someone is a reasonable thing to do. Of course it goes without saying that you ensure whoever gets it takes care of it. Better it be on display at a rich person's house where I might get an invite to see it than in the basement where nobody can see it.
As has been noted museums tend to be on the brink of financial insolvency. Getting a rich person to rent some art is a valid way to get enough money to stay solvent and continue to provide art to others. There is a lot more great art worth preserving than there are museums in the world.
Note, the above is not a comment on IF it is done. I have no idea if it is possible for someone to actually do this.
If you constantly collect items, and never discard them, then your collection will just keep growing (as will your warehousing bills). There's far more than can be put on display at many locations.
What else would a museum do with a piece on loan to them that they don't want anymore besides return it to the owner? That could be another museum or a private collection.
As has been noted museums tend to be on the brink of financial insolvency. Getting a rich person to rent some art is a valid way to get enough money to stay solvent and continue to provide art to others. There is a lot more great art worth preserving than there are museums in the world.
Note, the above is not a comment on IF it is done. I have no idea if it is possible for someone to actually do this.