I've added the same webmap multiple times under Data and configured the visibility of their layers in different ways. Unfortunately, this does not work if the layers are not visible on the original webmap. They must all be turned on, which makes a mess of a webmap if used anywhere else.
Please allow the visibility toggle in Experience Builder on a webmap's layers to override the visibility of the layer in the original webmap. I put in a ticket with Support and they were able to confirm the same unintuitive behavior on their end.
