When updating the URLs of services in a web map, it doesn't indicate which of those sources are direct AGS services and which are wrapped in an AGO layer item. If it's the latter, changing the layer's URL in AGO Assistant has no effect. You must check the web map's AGO item description page to see which layers are actually pointing at AGO layer items, then update the source of the layer item to point at the new REST URL.
I understand it's probably a bad idea for AGO Assistant to update the source of AGO layer items without making it clear what's happening. Which is exactly why it should indicate if a particular web map layer's source is an AGO layer item and either a) make it uneditable, or b) change the source of the AGO layer item and explicitly warn that the change will affect any other web maps using the AGO layer item.
Of course, I'd prefer option B, but option A would also be better than the current situation.
Separate but closely related issue: If a web map layer is using an AGO layer item, it's currently not possible to change its source to a different AGO layer item or to a direct AGS service. You have to create a new layer in the web map itself, and delete the old layer.