Without paying the athletes. Nice.
Though it is not a perfect system, big time college football programs "pay" their students room, board and tuition, worth at least $40k/year, which would be similar to minor league professional baseball. Considering a college grad in the US makes over $2 million more during their lifetime than a non-grad, for the vast majority of college athletes who never go pro, it is a pretty good deal.
It's possible, but universities are the only viable infrastructure for such an attraction. The school I went to has a 60,000 seat stadium for a university of only 10,000 students in a town of only 10,000 people. Outside a university setting, no one would be able to afford the stadium, coaches, equipment, medical staff, travel costs, etc.
They exist, but they're not followed much more than children's teams.
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Personally, I love high school and elementary school sports. In fact, I generally prefer them to professional teams. You need to remember only a small percentage of colleges in the US have major athletic programs. Most have programs that are probably similar to what you are use to in New Zealand. In the state in which I live, we have 10-15 colleges/universities, but only 2-3 with big time athletic programs.