Ooh @Mama2Drama theater is a great hook! At many boarding schools with after-school sports requirements, participation in theater can replace the sport, which can free up a lot of time. As you rightly surmised, theater IS considered to be enough of a group activity that requires the same collaborative commitment as sports.
You are also correct that arts boarding schools like Walnut Hill, Interlochen, Idyllwild, and the like, while providing a state/federally mandated high school curriculum, do not have college prep as their primary focus. They are generally for students interested in conservatory/art school/professional preparation. And, unlike most prep schools, it’s not unusual for students to only attend such schools for their last two years of high school. Some do start earlier, but it’s fairly common for students to start at arts schools in their junior year. At traditional prep schools, it’s much rarer; the majority of students start in 9th or 10th grade.
By the way, @Mama2Drama, GG was also young when she applied (she was 12, and entered BS in 9th grade a few months after turning 13. She will graduate this June, just after she turns 17). She did not take CC classes, because our local CC could not offer her the academic rigor we could homeschooling, though it is a very popular option for many homeschoolers. Had we chosen a hybrid route, we’d have had her do lab science at the local high school, because we live in a state that is supportive of homeschooling and most districts have good coordination with homeschoolers, if parents are willing to be flexible as well. We found good online courses and cobbled together appropriate curricula. Most of our homeschooling was EC’s 
Young students may find their age a barrier at some schools, and it was clear when we visited some that they simply considered 13 too young to be at boarding school. But others had no problems whatsoever, and none had an issue with homeschooling per se, though they were quite thorough in asking about our philosophy and approach to homeschooling. In case it helps allay your concerns, GG applied to eight schools and was accepted at four, waitlisted at two.