It is really hard Fleur, but unless children are so unbelievably able to hit success first time, each time on everything they do, then it's also really normal to see them upset. I see what you mean about sport, but it so depends on what they do, and to what level.
For example contact/team sports (like football, netball etc) you can experience the black and white win/lose analogy that you use. Where it can get more grey is when the win/loss is filters down to the striker/shooter/goalie as examples. There they are more exposed, and the risks higher to feeling success/failure as an individual vs the collective team. Even though they are in a win/lose sport. Disappointment is disappointment.
Picking an individual sport such as swimming, or athletics as examples, is a little more complex, as they are competing more against themselves to strive improvement against their own ability, and does have similarities to music advancement. One can attend a swimming or athletics meet, win all the heats, races, competition...and still lose. Because they didn't beat their personal bests. No amount of medal collecting on the day overcomes this. Or they came first in their age, but last in the race (because it has put children in by their seed times, not age). And age isn't relevant for all sports. So winners can still feel loss because they came in 10 secs slower than their last time), and those that lost, can feel great (because they took 10 secs off their last time). it's all relative to where they are at in their development.
What is the same is that these type of sports focus on the same thing. It's the effort, the attitude, the attendance, and work done in training. like music has with practise (excluding the flukes that can do both without work). The exam, or meet, is the "opportunity" to reflect of this. BUT, like with anything, the nerves, injury, dodgy piano...or whatever the curve ball of the day is, they can produce a result that is not reflective of their hard work. Shit happens, but a good coach, like a good music teacher will not focus or dwell on this, but will move on, and encourage the child to learn from it, as it happens to the best of the best. All the time.
Btw swimming is also a sport that many exceed to high level, but hate/don't compete. Just like musicians who do not feel it necessary to have an examiner determine whether they are good or bad. The benchmark should be against your own, as there's only a short window where you can assess performance against, say age, physique, years spent learning instrument/sport etc. At some point it all becomes a level playing field. The 55 yr old chess player losing to a 10 year old knows that feeling well 
We've sat many exams now across violin, piano and voice, and some of the results are bang on, and others have been as a result as a cock up or 5, and it's hugely frustrating. But we talk about how much easier that is to stomach than the child who's frozen mid performance, taken themselves off stage and not returned. Yet they are brilliant normally. This is for us, I know not all children can brush off so easily.
But I try and get DD to battle through all the things she does as the development curve that includes maturity is the same. Sometimes I am not very successful but I endeavour to keep trying!