I am very surprised guys under contract to be teammates in future seasons would do this. Not just because of professional etiquette and team solidarity, but because they depend on each other out on the track and their relationship also puts strain on all the people who support them from mechanics and strategists to public relations and principals. I can't see even the world's worst diva getting away with that without the team going absolutely nuclear on them, for all the reasons people have listed on this thread.
Is it really just a matter of rich, pretty 20-somethings having the runaway ego and self-defeating irresponsibility of celebrities or Republicans?
Auto racing, and particularly F1, always has had this interesting dichotomy of individual sport vs. team sport.
What you state is true but so is the desire of these Type A hugely successful athletes wanting to be a world champion. Everyone remembers that, not which team won the title. Plus, rule number one in racing is beat your teammate. Because that is the only apples vs. apples comparison you can ever have since all the other teams are so different, and again in F1 have completely different cars. So, the only way you can truly be measured is how you did against your teammate. Thus, by default, it automatically creates discord between teammates.
Think QB or goaltending controversies that occur. Sometimes, the athletes think team first and don't make waves. Other times, they think about themselves first and cause the team's harmony to be wrecked.
F1 has a long history of teammate controversies. Look up Senna vs. Prost when they were teammates. Heck, they crashed on purpose in the final race in Japan to decide the championship. Look at Hamilton purposely backing up Rosberg hoping someone else would pass him to take away crucial points that would have given the title to Hamilton at Abu Dhabi despite the team telling Lewis to absolutely stop it. Multi-21 between Vettel and Webber. Alonso staying in the pits to screw Hamilton from getting new tires in time during qualifying when they were at McLaren.
Once, when Bernie Ecclestone owned Brabham, in the last race of the season, he told the second driver to move over if necessary to let Piquet by since he was fighting for a title. The second driver said he absolutely won't do that. Bernie said, "So, we made sure it wasn't a problem. We didn't give him enough fuel to finish the race."
Enzo Ferrari loved to pit his drivers against each other to "motivate" them. Some say that was the reason which caused Wolfgang von Trips, under intense pressure, to fatally crash at Monza handing the championship to his teammate Phil Hill in 1961. (Ironically, Hill and von Trips got along just fine.)
Then, there are the times teammates are told to support the other driver with the promise it will be their year the next season. Many argue Ronnie Peterson was a better driver than Mario Andretti, but Peterson duly supported Andretti to the title in 1978 as asked, because he was told it would be his year to go for it the next season. Sadly, Peterson died at the end of that season.*
Gilles Villeneuve also played the dutiful teammate, supporting Jody Scheckter who won the title in 1979. Again, Gilles sadly never had a chance to have his turn, as Ferrari took a downturn and then he was killed in 1982.
* - The stories of the only two Americans to win the World Championship are tragically coincidental. As the season wound down, the only drivers who had a mathematical chance of beating them were their teammates. In both cases, their teammates were killed, clinching the title for the Americans. Both fatal crashes were at Monza.