I think it's a mistake to evaluate sport by only looking at the very highest level. Almost nobody competes there, and in professional sports, I'm not sure that's always where you would find the best role models either :-)
Also, I think you have touched on 3 different things - military training, martial arts training, and sport. (The MAs that we practice today are no longer military, in general - IMO - although they may have that in their heritage somewhere.)
I believe that classing these pursuits as "higher" and "lower" is misleading.
I don't believe any of them necessarily make you a better (or worse) person, although they all have that potential.
Actually, I would classify them as higher to lower.
Let's take the middle one, military. The kind of training you can receive is hand-to-hand or close quarters combatives. You can get training in rifles and pistols and their tactical usages. You can get training on military tactics and strategies. Training with machine guns, rocket launchers, etc, etc, etc. Sound familiar? If not, then just replace the rifle with the sword. Replace all the current weapons with older weapons, like spear, naginata, bow, etc. Basically, it's a newer version of Japanese koryu.
The differences, however, are greater. In the military, you aren't training for the betterment of yourself, but rather for the overall organization. Your life for the U.S. In the martial arts, quite a bit of your training is about *you*. The military only cares about you in regards to you being a part of a unit, part of a command, and part of a force to use.
In terms of the betterment of *you*, the military falls far short of the martial arts.
Now, let's jump to the lowest on the list: sports. There is an overinflated sense of self that is not only trained, but bolstered. From the Olympics down to little league tee-ball. It's about winning and losing. Sportsmanship means either he has a strong striving spirit to win or that he is a gracious loser. No one wants to be the person who drops the ball and lets the other team win. That person's emotional state is not very good.
Course, I lump the McDojo Karate/Taekwondo school that has its window full of trophies in the same category as sports. But, any martial art worth its name not only teaches you to be "strong" (budo strong, not physically strong), but also has the same notions that Ueshiba had. Ueshiba embodied these ideals of strength, harmony, and self-betterment. From the few bits and pieces I've read, Kodo had it, too. I think Chuck Norris is quoted about how the ideal is to be so good that you don't *have* to fight. Might have been someone else, though.
No matter what you're doing, you train to have that base of "power" or being "strong" (again, these are from a budo definition) so that you have the options and opportunities of choosing harmony. That's what builds the better *you*.
You won't find that in the military or sports. In fact, it is directly opposed to sports ideals. You might have a chance of finding it in the military, but orders from superiors will over-rule it.
Still, after all that, none of it means that you can't have a form of "competition" in the martial arts. As Peter Goldsbury noted, Ueshiba took on challengers. There was a winner and a loser in them. Tenryu could not get the better of him. IMO, it was the aiki that had changed Ueshiba so that he didn't view these people as competitors. As Ueshiba said, "I am aiki!". Not the "win-win" view of modern day defined "aiki" or the harmony of joining view of modern day "aiki", but the complete body skill aiki that he learned from Takeda.
IMO, the aiki skills changed Ueshiba so that he did not need any form of competition because he was at a place of being "strong". Competition hindered his martial progression because it focused too much on winning and by doing so, the focus on aiki skills was diminished. Diminishing aiki skills equates to not being "strong" which then degrades the ego when coming into contact with other people (in this case, this means all the challengers). Degraded ego means aiki skills not functioning well because the mind isn't "pure". Vicious cycle that feeds upon itself. Quit focusing on competition and start focusing on building aiki skills and all that goes away.
One other factor brought to my attention is that building aiki skills in oneself, possibly causes a change in one's demeanor. The better you get at aiki skills, the more "strong" you get, the less you view the world in a fearful manner.
Competition? In certain defining terms, yeah. In a sports-like venue, no.