It might be useful to distinguish between subjective aesthetic factors such as whether the performance had a requisite flair or elan, and functional aesthetic factors such as whether the gymnast stuck his landing or the diver made a big splash in the water. The latter qualities, though aesthetic, are fairly easy to make objective, and you could conceivably program a computer to make such judgments. Women’s gymnastics has too much dancing to qualify as a sport, but could easily be reformed (though there’s no sign whatsoever that gymnastics fans want such changes) to be more objective. The awarding of points for subjective aesthetic factors is the dealbreaker, not the mere presence of judges; again, men’s gymnastics is (to me) indisputably a sport.

Contrary to the assertion in the OP, subjective aesthetic judgments are ubiquitous in basketball and football. Computers or robots will never replace the officials in those sports, because a computer is incapable of distinguishing between the four steps taken by Michael Jordan making a layup, and the same four steps taken by someone less graceful or even less famous. Similar considerations invariably obtain in football, though the considerations are aggregate rather than individual–certain calls are made or not made based, not on whether the foul occurred, but whether the beauty and flow of the game would be enhanced or diminished by throwing the flag Of course basketball and football are sports, just that the aspect of subjectivity is a little more complicated than you are allowing.

Probably a good rule of thumb!