If you look at your post then you can see that your school situation is unique and not really the norm. Many do not have alternative education choices like Montessori or they are too expensive. You are very lucky to have a wonderful option for your child.
Some choose to homeschool because they don't have these options and they can provide a better education themselves. Or their schools may be dangerous, run down, overcrowded, etc. Or they've had bad experiences with teachers that will not change.
But even with alternative ed choices some people choose to homeschool because they want to impart not only their values on their children but they also want to guide their education in a way that meets their needs best.
Many children are not at 'grade level" for all subjects yet most schools are set up to teach only that way. So kids are told to stop reading ahead or labeled a troublemaker because they are bored since work is too easy for them. Homeschooling eliminates these issues since the child can work at their level and learn about things that interest them for as long as they want-not just a semester or 50 min school period.
There's an excellent book called "Dumbing us Down" by a teacher, John Taylor Gatto, that also explains in depth how the average student is taught to basically be a "cog in the wheel of society." Many homeschoolers want to fight against that and teach their child to be independent and critical thinkers. The realities outlined in this book are really some of the primary reasons I homeschool.
I'm not trying to get into a debate with anyone why homeschooling is better or worse than public/private schools. It's all relative to the individuals. Parents do what they think is best for their child.
This is just one homeschooler's perspective.
I chose to hs my dd10 because he just wasn't at grade level for all subjects. He's at grade level for math, but his reading skills are below grade level. He was falling more and more behind because the class was targeted towards the grade level, not his level.
I've been homeschooling him for about a semester. I actually went back to about second grade level for reading, just to bring him up to speed. He's caught up and is now reading 4-5 grade level books! We're also improving our writing skills. OUr routine consists of lessons in the morning, lunch break, then an "outing" every afternoon. Sometimes its the library, museums, etc., sometimes its just a shopping trip, but we're always looking for opporutnities to learn. For example, I have him help me figure out what is the
best buy based on price and weight. We also write the shopping list, which gives him practice in writing. He's also in various structured activities, such as scouts, church youth group & Sunday School, gym, art classes, etc. His days (and mine!) are full.
He's quickly catching up, now that the anxiety is eliminated. He was so out of sync at school that he had all but given up. I intend to re introduce him to public school at the middle school level, but I will not send him back to his current elementary school. There was simply way too much anxiety associated with that school for him, and the school had developed a negative attitude towards him that will never change (oh, I've tried everything, once a mind is set, its impossible to change it!)
The change in ds has been quite heartening! He's a willing pupil and fast learner. Also, one of the advantages of homeschooling is that the child can learn for the sake of learning, not to meet some picky requirements. Case in point--my dd9 had to do a social studies project last Fall. They were studying the Spanish missions of Texas (we live in Texas). Well, part of the project consisted of building a mission. There were so many picky little requirements she actually froze, she didn't have any idea how to proceed. We helped her and almost ended up doing it for her. Yes, I know, what did she learn? The point wasn't to learn, it was to complete the project and get the best grade possible. That's the game.She experienced so much frustration and tears from what could have been a learning experience. A few weeks later, ds decided he wanted to make a Spanish mission. He made it 100% on his own and did a fantastic job. It engaged him for hours, and he spent a lot of time studying books, looking up facts on the internet, in essence, a learning experience for him. His mission was fantastic! I heard dd tell ds, well, it was fun for you, you didn't have to worry about getting it "right". Her teacher picks her projects to death to the point she is discouraged from even trying (took of points for too much white space on pictures she cut out of magazines--about 1/32 inch!, took off points for glue run over--about the thickness of a single thread!--shes in fourth grade, not a professional artist!). On the other hand, ds was able to involve himself in the project and all it involved without the fear of not pleasing a teacher who thinks teaching equates to picking.
Yesterday we were at the library and ds spied some books about the Transcontinential Railroad. We'd discussed it some as part of a story he read. He brought home several books and now wants to make a model of the railroad, which I'm sure will be a great learning experience. Once the anxiety is eliminated, he can concentrate on actual learning. By the time he reaches middle school he might even be a grade level advanced for his age! I'm just glad I have the time and resurces to homeschool. Its not for everyone, but its the best solution for our circumstances.