Sure looked to me like the kid could be called for a balk on that play.
I'm assuming TN HS baseball uses the NFHS Rules Book. Rule 6, Sec. 1, Art. 3 reads:
For the set position, the pitcher shall have the ball in either his gloved hand or his pitching hand. His pitching hand shall be down at his side or behind his back. Before starting his delivery, he shall stand with his entire non-pivot foot in front of a line extending through the front edge of the pitcher's plate and with his entire pivot foot in contact with or directly in front of the pitcher's plate. He shall go to the set position without interruption and in one continuous motion. He shall come to a complete and discernible stop (a change of direction is not considered an acceptable stop) with the ball in both hands in front of the body and his glove at or below his chin. Natural preliminary motions such as only one stretch may be made. During these preliminary motions and during the set position until a delivery motion occurs, the pitcher may turn on his pivot foot or lift it in a jump turn to step with the non-pivot foot toward a base while throwing or feinting as outlined in 6-2-4 and 2-28-5, or he may lift his pivot foot in a step backward off the pitcher's plate which must be in or partially within the 24-inch length of the pitcher's plate. In order to change to the wind-up position, he must first step clearly backward off the pitcher's plate with his pivot foot first. After the pitcher has placed his pivot foot on the ground clearly behind the plate, he then has the right to throw or feint to a base the same as that of any other infielder.
I can see the umpire explaining the pitcher didn't come to a complete stop and during his 'quick' stop the glove was positioned above his chin. Kind of a questionable call on the play but not necessarily a wrong call. Why was the pitcher throwing from stretch on the play as the video seems to show there wasn't a runner occupying 1st base. When I pitched, I was taught to always pitch from a windup with a runner on third and first base unoccupied as you're more likely to be called for a balk pitching from the stretch. I was also taught that with a runner on third and pitching from the stretch, make sure you come to a complete stop and count to at least one to freeze the runner(s) and to lessen the possibility of a questionable balk call made by an umpire.
Ticky tack borderline call on the play? Absolutely but not necessarily the WRONG call. I've seen some 'better' umpires let it go and then, in between the next half inning of play, seek out the team's head coach and tell him to make sure his pitcher comes to a complete stop as 'you don't want to lose this game due to a balk.'
Also from the reactions after the play, this call didn't end the game. (Edited to add: looked it up and this play occurred in the bottom of the 4th inning.) Tough way to lose a game and a tough way to end the season & some players HS baseball careers but "you can't win if you don't score" and Clarksville didn't score a run in the game.
I saw a 'phantom' balk call end an extra inning game and a team's season two years ago and that balk call was completely bogus as the HP umpire commented to me before the half inning started that he needed the game to end soon as he had to be at work in 6 hours and he hadn't yet slept.