I would say it is more embarrassing (and sad) for the POTUS to tweet this out to everyone.  This coupled with his constant verbal diatribes against the media and others can be a problem.  

Look what happened to Senator Scalise a couple of weeks ago with all the crazy rhetoric going on.  You never know when some idiot will snap and attack a journalist (like when the Guardian reporter was body slammed by the state Republican candidate recently) or something worse.  

Threats are threats - even if against Trump.  You are kind of proving my point with your link.   

I think the criticism is warranted since not everyone has the same mind set as you or I (we see the levity in some of this - some do not).