'Tornado Emergency' is enhanced text added to a Tornado Warning designed to get your attention similar to the 'Particularly Dangerous Situation' text added to some Tornado Watches.
Should weather alerts be posted here? No. Not as a general rule. Not unless it is causing public safety agencies to go into active response modes.
I see the alerts section here as something to draw attention from those far removed from a local area, not as something to warn local residents of imminent danger. I might be interested in logging on to Detroit if they have a tanker fire or an airliner crash for example. Weather alerts don't mean anything to me anywhere other than my home area. Also I have a number of other products to monitor weather conditions nationwide. I wouldn't come to RR for that.
Now, if there were an active storm doing destruction and the emergency responders were in rescue and recovery mode, being dispatched, giving locations affected, requesting mutual aid and so on, that might be different. The Greensburg, KS tornado a few years back or the Moore, OK disaster would be good examples of things to post as alerts.
I don't know, maybe it should be called something other than an alert for this board's purposes.