lol - both runners forgot how many outs there were. They had to have thought the catch in CF was the 3rd out or they'd have held to tag. That he just trotted home is a secondary consideration - "Hey, maybe he'll drop it, and it's a free run." The announcers are correct. By failing to tag up, the runner at 3rd is automatically out by touching home plate out of order. However, a more correct call for out #3 is probably the runner at 1B who also didn't tag up touching 2B out of order before then.
I think that's what the umpire was trying (and failing) to explain: the "timing" part of the play is which runner needed to be called out first. At that point, it's kinda academic, since they're both out. No real need to send it to New York. Just call them out at both bases and let the scorer figure out which one to write down as official.
EDIT: Actually, thinking on this further, it was probably just the runner at 3B who forgot the number of outs. The baserunning play on an iffy deep fly ball like that is to go halfway. If the fielder misses or drops it, keep going. The runner at 3B sees the catch, thinks it's out #3, and continues on to home. The runner at 1B, already halfway 2B, takes his cue from watching the lead runner. "Oh - must have been dropped out there." Then he accelerates to try and get the extra 90 on the throw in. That entire play had to confuse the hell out of the 3B coach and the trail runner. Runner is like "What do I do? Slide? Stay up?" while the coach is staring at the idiot who decided just keep trotting home because why the hell not? The dugout's that direction anyway.