Let me preface this with saying this was my philosophy for raising my kids, so it may or may not mesh with your own. Just sharing what's worked for us.
What I wanted most for my kids is that they be well balanced. I didn't care if they had the best grades, were the best athlete, or had first chair in the band. What I wanted is to allow our kids to explore all aspects of learning and creativity, so we made two rules with all of our kids:
1. When you start school, you add one sport and one musical instrument to the mix. You can try whatever you want, but you can't just quit. If you decide you don't like one sport, you need to find something else to replace it. If you decide you want to drop one instrument, you need to find another.
2. You may only participate in one team sport at a time. You may only take lessons for one musical instrument at a time.
By the way, both of my older kids ended up being nationally ranked in the sport they chose, and because of their dedication and commitment we definitely did the whole drama thing - throwing up before competitions from nerves, meltdowns afterwards. But going through those experiences and learning how to draw somewhere deep down to get through the nerves, to give everything and find out that sometimes it was enough and sometimes it wasn't - there were lessons they learned in sports that I don't think they could have learned in academics.
My son did have to quit band in high school - not because he wanted to but because the band director made him quit when he missed a practice for a sports practice.
You're right that most kids don't get scholarships for sports, but the lessons they learn are still quite beneficial. Plus, getting them to engage in physical activity on a daily basis is such an important habit for a healthy adulthood.
Oh, and I did refuse to let my youngest join soccer when I found out it required us to commit 6 days a week to it. There were other sports that could teach the same lesson without eating up our entire lives. He's picked swimming and is loving it.
