Having fun reading the variety of parenting approaches here... everything from supporting a dozen+ full-time activities to letting 'em run wild. Each have their merits, but we went for a balance of the two.
My parents were certainly from the let 'em run wild camp. As kids, all our sports took place in front yards, streets, fields, and ponds. I did play a season of organized little league, but only because I could get to my games on my bike. My parents never saw a single game. But sports ended for me by the time I was about 12. That's when I literally started running wild.
I didn't get back to sports until my son was 6 and we signed him up for soccer. I just watched his games that first season, but got sucked in to helping coach the next year. I then coached his soccer, baseball, and hockey teams for the next 10 years. I starting playing softball during that time, then hockey, then snowboarding. With a demanding career, my son's sports and daughter's dance, I never had time for golf. That didn't start until my kids started college.
Even though we were much more involved in our children's activities that my parents were, we limited their participation so they had some "running wild" time too. We weren't big fans of over-committing our kids or giving them too much structure. We were also never under any illusion that our kids would make careers out of their "recreational" activities, so we made sure they had plenty of time to just be kids. Although my son is an incredible hockey player, we limited his travel team participation since it involved weekly 3-4 hour trips to LA or SF, as did my daughter's dance competitions.
They turned out great, recently graduating from Cal Poly and UCSB, and still play hockey and dance for enjoyment. I have no regrets about putting off golf until I was 46. I'm now playing up to 3 times per week and keep right up with everyone who started much earlier.
Although my adult children now have busy schedules, they occasionally join me for a round of golf. My son has now played 8-10 rounds and shoots in the 90's using a hybrid, a few irons, and a putter. My daughter just played our new 12-hole par 3 course with me using only my hybrid and putter (she hit 3 greens). I'd love to have played with them while they were growing up, but seriously, how many kids prefer playing sports with their parents when they could be out playing with their friends!?! I'm just happy they do now.