Yes online school is not at all the same as homeschooling. However, online classes can be part of homeschooling. There are many ways to homeschool and many possible ways for students to learn. It is not necessary for the parent to teach every subject to the student, and in some cases the parent is not ever the direct instructor.
Since you are at least considering homeschooling, I will offer my general thoughts having homeschooled two gymnasts for several years (but not yet HS.) I am not at all trying to be all rah rah homeschooling. Homeschooling is not the right choice for every family or student, just as “school schooling” and online schooling are not the best choices for every situation. Just offering some info.
The major benefit of homeschooling from a time management perspective is the student can eliminate the busywork and still learn. Learning does not require that a student produce a mountain of work. There are of course many skills that require repetition to master and when learning, it is often the case that writing about, drawing a picture about, making a movie about etc. the material one is learning helps the student absorb the material. And these processes generate work product. The problem with regular school (including online school) is that the work product expected frequently ends up being so much, it overwhelms and could even undermine the process of actual learning.
Also it is individual how much work product a student needs to do for any particular subject or skill in order to grasp the subject or become facile with the skills. Homeschooling allows a student to adjust work production accordingly.
Based on the homeschooling families I know who have homeschooled HS students, homeschooled high school students overall tend to become more self directed in their studies. Obviously this varies student to student. A parent certainly needs to be sure their child is learning what they are needing to learn, (no matter how the child is being educated.) But while the parent could be their child’s primary instructor, in many cases, this is not needed. ( This is aside the assumption some make that being a primary instructor is not possible for a parent who is not a credentialed or trained teacher. For most homeschooling families, the parent is not an accredited or trained teacher. And what I have been told by my friends who are credentialed and trained teachers and who homeschool, the skills needed for teaching in a classroom do not help much when teaching one's own kids.)
Depending on where you live, there may be classes aimed at homeschoolers, (including both online and in person courses) community college courses (that may fulfill both HS and college credit at once) etc. Also it is amazing how much instruction, resources, tests, syllabi, etc. are available entirely free, online, and of course curriculums and text books can be purchased.
There may be more than one way to legally homeschool as well. (And in some states it is very hard to legally homeschool.) It all depends on what state you live in. If your son is looking at college gymnastics, I also suggest call the NCAA as I have heard that they will offer parents information about what NCAA compliant homeschool paths are available in their state.
It can be exhausting looking at all the possibilities and figuring out what works best. The best advice I got when I started homeschooling my sons was that “if it is not working, you can always try something else.”