In our state, homeschooled kids DO have the option of doing those things. But I'd much rather have a child who didn't go to a prom, but did go to Yale, instead of a kid who got to be homecoming queen and can't write a simple essay because no one bothered to teach her. My husband, my friends and I all value our educations far more highly than any opportunities for dancing or cheering we had in public schools, which leads me to believe those opportunities really won't be missed.
I also value education, and I certainly don't think that going to prom and going to Yale are mutually exclusive. I know several people who have done both, and I want the same thing for any children I may have. I really don't get the implication that the public education system produces individuals who are never taught to write a simple essay. Public school is what one makes of it. My high school was named in an ACLU lawsuit as an underfunded, "ghetto" school, but the fact is, I got a decent education. At least my high school education allowed me to graduate from the UC system in 3 years, Phi Beta Kappa, Magna Cum Laude. This in turn got me into every PhD program I applied to...AND I got to do homecoming floats! My life didn't revolve around such things, but in my case, the opportunities would have been missed.
I really don't know whether home schooled kids are allowed to participate in these activities in my state. Perhaps they are, but I have never met any who did.
I can only really base my opinions on what I've seen of my own high school (in a different state), and my brothers' and friends' schools, and my experiences in college in this state. My brothers who went to school here, and most of the people I went to college with, couldn't write an essay. Sure, they got good grades, but the curriculum was weak on basics, and they would not have been prepared for a top-of-the-line school. So from my experiences with this state's public education, homecoming and Yale would seem to be mutually exclusive.
I'm not trying to say that anyone who wouldn't choose to homeschool doesn't care about education, because obviously that's a ridiculous assertion. I was simply trying to correct the view that homeschooled kids aren't ever socialized, or that the typical high school activities are important to all kids. All I remember about high school is a lot of awkwardness and pain - and my high school had no cliques and no real bullies. It wouldn't have fazed me or any of my friends and family to have foregone the typical high school experience so we could learn at our own pace, on our own time, out in the world and at home in books.
Some people do well with homeschooling, and some do not, but to dismiss an educational choice out of hand because of a preconceived idea that is flat-out wrong isn't helpful to the OP or to the discussion in general.