Let me add my 2 cents with our Troop's recent experiences.
I've been in our Troop just over two years and recently was asked to be Scoutmaster, the position I now hold. Since joining the Troop, we had not competed in a camporee until just a few weeks ago. During that time the prior CC and SM debated the value of camporee; that plus scheduling problems led to us missing camporee for those two years.
I also recall a discussion at my first summer camp with one of the other leaders about the merits of competition. His view was basically what I've read others say on this thread, "Why does everything have to be a competition?"
For these two years, the patrol method and boy leadership really declined.
This year, the new CC and I decided to make camporee a priority. The Troop spent about 3 months preparing including running our own pre-camporee training event about a month before the real one. It was a great event, but one patrol really struggled, not only with the skills, but more so with moral and teamwork. Out of our 9 patrols (yes, a big Troop!), they finished near the bottom.
Then comes the real, council-wide, camporee. What a great weekend. It started with the Scouts doing all the work to setup camp Friday night (yes, the adult leaders gave a lot of early guidance, but the Scouts really did it all). The result was that several of our 9 patrols received perfect scores on campsite inspection the next day. Saturday's competitions were also great. Many patrols also received perfect scores at the various events. At the end of the day our DE stopped by to inform me that one of our patrols received top score for our district. You guessed it, the same patrol that finished near the bottom of our training event! You should have seen the look on their face when the announcement was made at the Saturday evening event in front of 2,000+ Scouts, Scouters and parents! The irony is that the patrol leader is son of the leader that questioned the merits of competition nearly two years ago at summer camp!
Do you think that did wonders for that PL and his patrol? You betcha. And how about the entire Troop? Well, they couldn't have been happier and prouder. Even Scouts from patrols that didn't score well and struggled through out the day were excited and celebrating as we returned to camp that night. After returning to camp (and at our next meeting), we recognized the great achievement of that patrol but then also stressed that it was a team effort of the whole Troop that really made the difference. Had everyone not chipped in to setup camp, camp inspection scores would have been lower. If only a few had gone to our training campout, our patrols would have been less prepared.
OK, what if we hadn't taken first in our district? Well, there were plenty of positive things that came from the experience. The trick to making competition a rewarding experience despite "not winning" is to find those positive things and build on them. Find things that need improvement and make it fun to improve on them. Turn negatives into positive. Celebrate the victories, no matter how small. If the attitude is that there is only one winner and everyone else is a loser, then there is a flaw in the competition. Everyone should be able to "win" in some form or another.