If ball hit in the corner by a fast runner and the RF picks it up and the Batter tries for 3rd is the most likely scenario for a 2B making a throw as described. That is probably all of a 300 foot throw and would require a cutoff.
On a base hit with runner going 1st to 3rd or a runner tagging from 2nd should never happen unless RF made a mistake and threw the ball to the 2nd baseman who is likely to have been the closest fielder to him from the infield. Although in these scenarios you can see why he would be "rushed".
I think the question really being asked by IAMOM is: Is it a legit move for a coach to ask players to speak up as leaders and then 2nd guess them publicly when they do?
That bit of distinction matters here I think. If you want 15-18 year old kids to "lead" each other, then you get what 15-18 kids are which can range from Dumbass 1st degree to Valedictorian. They can learn and improve with experience and as the teachable moments come. But to think you aren't going to get some junk in the mix seems shortsighted.
IMO A better move for the coach would have been to express his opinion of what the catcher should say and how - at the next practice quietly with the catcher 1-1. Then encourage him to keep being that guy - if that is what he wants.