If you look at records in places like Philadelphia and Chester County, PA, which had a large influx of Welsh people in the late 1600’s/early 1700’s you’ll find that the Welsh ‘Rhys’ frequently and quickly became Rice, Reese, and Reece in America. It didn’t vanish, just changed to an English spelling. Many Welsh people were using patronymics when they arrived in America, with names like “Davis ap Rhys.” The patronymic either got dropped so that the children of the man named Davis acquired a surname of Davis, or converted so that “David ap Rhys” became “David Reese,” and his children acquired a surname like “Reese.”