I'd also like to see a less competitive competition. The example that comes to mind is randonneur style bike riding.
I'm of two minds about this. (And my experience is mostly in IDPA these days.) On the one hand, yes, as competitors get closer to the tops of their classes/divisions, they do seem to notice scores more keenly and there is a bit more friendly striving to best your closest rival-pal.
But for the majority of the shooters who won't be taking home a plaque, there seems to be very much a sense of shared accomplishment (an
esprit de squad if you will), which I think must be similar to the randonneur ride you describe.
We set out together in the morning and with great encouragement, fellowship, and humor, we faced our six, twelve, sixteen or however many challenges/threats as we came to them in our grand march around the range bays. Then, in the afternoon we gather around the clubhouse, pass out some plaques for those who happened to come in with the fastest times, eat some pizza, and swap tales of the little victories and hilarities that occurred from the adventure we've completed together. Walking through the clubhouse after a match, the place is always abuzz with guys retelling how they handled the sudden pop-out target on stage 6, or pantomiming how they took the cover-shot on stage 10 from some goofy position, or jokingly lambasting the M.D. for coming up with the evil whatzit on stage 2 that was so tricky -- all to gales of appreciative laughter from their pals who shared the experience. Veterans all, of a shared campaign to face down the flat cardboard enemy.
I can see how there is less of this in something like bullseye or highpower where the targets are identical and designed to demonstrate a pure, distilled skill, and the matches are solitary, focused, introspective events. But the scenario-based "action" shooting sports provide a totally different vibe, and I think that's why they've become so popular.