[dragonwriter]

> When I say "small-scale skirmishes", I'm talking about relatively small border conflicts and the like that last less than a year and kill tens to hundreds of people.

Football skirmishes that kill people on that order aren't unheard of, and I know of no evidence even suggesting that the popularity of professional sports even correlates with (much less causes) a reduction the kind of small-scale skirmishes you discuss (even before considering whether the degree of such effect is offset by violence directly associated with sport.)