Private collections of billionaires would shock you.
Items naturally go to the highest bidders. What you can see in museums is often on a temporary loan privately or from another museum.
I could not say whose collection is larger, billionaires or all museums.

Questionable ownership is an ever-lasting tricky ethical conundrum.

10 years after the Nazi art theft many items ended up in fancy restaurants, not museums. Times are changing and slowly even the Smithsonian is repatriating some artifacts only after a big stink.
Museums that have obviously stolen art use methods to tie up an item until the true owners have died. Of course its a nasty business dealing with 'priceless' things.