What's wrong with this???
 

"...Then things got crazy with 3:48 left to play. The Leopards Kyle Baker headed a ball off a corner kick towards the net, but the Northwest defenders kicked it away in midair. Next thing everyone knew whistles were blowing and the Leopards side of the scoreboard had a “3″ appear. Soon after, the head referee conferred with the linesman on the north side of the field and overturned the call in the Indians favor. But like a 2000 election recount, Leopards Head Coach John Henkel managed to convince the referee to go with his previous decision giving the goal back to Louisville. The referee apologized to the Northwest players and told them that the call could not be overturned after it was made."

Disputed Louisville Leopards Goal Against the Northwest Indians - YouTube

First off, let's stipulate that this a bad camera angle for determining whether the ball crossed the goal line.

It does not appear the ball crossed the goal line before being cleared by the defense.

We cannot see the AR or the referee at the time of the apparent goal.

I can think of a few possibilities here.

1) The ball did completely cross the goal line and we just can't tell from this angle.

2) The ball did not cross the goal line and either
a) the AR anticipated the play and signaled with a flag up the ball crossing the goal line, or
b) the AR moved aggressively up the line to reestablish position which the referee took as a "good goal" signal.

How should it be handled?

In my pregame, I always tell my ARs that on balls that go across the goal line and immediately come back out that the first thing I want them to do is go straight up with the flag. Then, after I've whistled, to give the good goal signal by sprinting up the touchline.

I think the referee was correct to confer with the AR after the protests from the defense. He has the ability to remove the goal and change the restart (to a drop ball somewhere around the edge of the penalty area, in this instance) up until the point the next restart is taken.

Presumably, the HC argued that a goal once awarded cannot be reversed. This is simply incorrect.

Exactly!!! This is what is wrong with it!! It's obvious that the CR called this a goal on his own. Otherwise, when conferring with the AR on that side, they would have both been in agreement and that would have been the end of the story. However, the AR appears to have told the CR that the ball did not cross the line completely (if at all) and had the CR do the right thing by nullifying the goal. As you stated, it should have been a drop ball situation, or if one team was in clear possession at the time of the whistle, it would have been their indirect kick (i'm not going back to look at that now in the video).

To me, there is no other explanation of this than the CR not knowing the rules and over-executing his power.