I think it's unusual for the chief of staff to go on a trip, particularly on a trip this long. The chief of staff is usually more of a chief operating officer in the White House itself, and normally when your principal—whether it's the president himself or the head of Cabinet agency—goes abroad, you have his deputy and those folks staying behind to help manage operations in his absence.
For the Saudis, anyone is better than Barack Obama...Trump has a strongman persona. And that endears him to autocratic leaders in the Middle East.
The regional governments are so eager to have more active American engagement that they will overlook any slights they might otherwise perceive in the president's view of their religion.