Oh, and I think this is somewhat in distinction to, say, large publics. They will likely have club and IM too, but the level of ability and commitment required for club, and sometimes even IM, can be pretty high.
It is basically just a scale issue–an institution of 40000 kids isn’t likely to have 20 times as many club teams or IM leagues as an institution of 2000 kids. More, yes. 20 times more, very unlikely.
Obviously there is a lot of other stuff to consider when choosing a college, but this is one of those things where for the right kid, a smaller or at least midsize college might be a little better for them in terms of opportunities to participate in well-organized team sports.