And it probably would have been poor form for the US to send a ship bigger than those in Chile service to their parade. You don't want to show off the host unless you've got bad taste.
This is quite probably the reason, you can't have a visiting ship be the de-facto flagship by virtue of being the biggest. Large USN ships have visited Chile (often on trips through all of Latin America) before for exercises, my father having seen a Nimitz-Class once in the late 80s or early 90s, I'd have to ask him again. The locals would rent out small boats to get people in closer for a better view.