>"Whenever looted cultural property is found in this country, the United States government will do all it can to preserve heritage by returning such artifacts where they belong" So what is the definition of "looted" here? Museums are full of things that fit various definitions of that word.
Many institutions built their collections through a practice known as partage or division of finds. They would go to a government and offer their services as archeologists in exchange for a portion of their finds. While this practice is rare post-1970, there's not any serious debate about the legitimacy of cultural heritage exported under this regime.
Purchases from the antiquities markets are also considered legitimate, assuming the export was originally done with the permission of the originating country.
Where things really get questionable is when material culture was extracted by imperial power or as spoils of war. The Rosetta Stone is the most famous example. Also, you've got things like the Elgin Marbles that were taken through fraud and subterfuge a long time ago. These kinds of things are grandfathered in as a legal matter, but there's a moral argument for returning them anyway.