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).
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.
Loaned items are specifically loaned to be displayed for a length of time, it's not like borrowing a grass trimmer off of facebook.