I think you need to clarify what you mean by "Idle", here. Modal VIs, for example, are only modal while "running" in the execution system, and thus not "idle" in that sense, but they are modal while not actually executing, and thus "idle" in another sense. The historic terminology is unfortunately confusing. I assume you mean that properties like Modal should only act while the VI is actually executing.
An example would be a floating information window, one that is shown to the User but only called when necessary to update the information. The VI is thus mostly "idle", by your definition, yet needs many of those properties active.
Floating and Modal already only apply while running, so by "running" you must actually mean executing (ie. in the middle of a call). You want these properties to apply only when the VI is actually executing, correct? I think you are motivated by the problem of an open non-executing dialog, which would be solved by this. My example is a VI that is being shown while not executing.
Note that we cannot attach examples on the idea exchange...
I agree with drjdpowell, this has it's use cases.
Often it's assumed a UI VI has to be continuously running (while loop, event structure) but that's not required. You can (easily) make a VI that doesn't do anything and have another UI VI handle it's events. I use this pattern a lot, although the VIs usually live in a subpanel. But the same could be done for floating or modal VI. The dialog is just a facade...