Yes, in that situation local breakfast shows make more sense but I don't think it is that common in countries with multiple time zones, though in the case of post-show breaking news I think the US versions do go into live mode in the West.
Most US National breakfast programmes are tape delayed except in ET. For example, The Today Show generally only airs live in the Eastern Time Zone at 7 AM. It is shown on tape delay at 7 AM in the five other time zones located in the United States. If there is breaking news the day of the taping, The Today Show will broadcast a live West Coast edition. The live feed will play out for the first hour then return to the taped East Coast version. Local affiliates will be given two five minutes time periods during the hour to break to local news. These occur at :25 and :55. Generally, local weather updates are given during these times.
Most of the national breakfast shows are preceded by hours of local news broadcasts, some starting as early as 4am.
Breakfast Television (BT) on Citytv started of course as a local show on Citytv Toronto eventually as the network spread across Canada local versions were created, most continue to air after many of the City stations' dumped their local newscasts, the breakfast shows proved to be popular viewing.
I said what I said!