Why would the STC tell Austin to fair catch the ball in the first place? I can think of only three reasons. 1) The STC was told to do it, 2) Austin was expendable from the Rams because of his mishandling of punts and the STC and the Vikings were aware of that or 3) to save all of the time for the offense and let's face it, the Cowboys do not have great ST.
As far as Garrett handling this, who knows the real story about who told who what and when they were told but with Austin being a veteran PR, there's no way he fair catches that ball on his own.
Garrett using Austin as the scapegoat is the surest way to lose this team. HC's take hits for their players, that's part of their job, keep the heat off their guys.