I don’t know if this has been requested before, or if for some reason this isn’t feasible, but would it be possible to add an option to this that would set the object display mode for all viewports instead of just the one you’re working in?
it is not perfect because drop down but you can still use keyboard to get to the displaymode in the list… so press “R” to highlight Rendered and press “R” again to get to the next entry beginning with R if there should be any.