To homeschool or not to homeschool: that is the question, but the answer is not binary.
The research is absolutely clear: the kids who do best academically tend to have parents who are interested in their education and, specifically, read to their kids from an early age.
After that, everything is mostly details. You will find kids who do well academically in public schools, in private schools, in charter schools, in homeschools. And you will find kids who are pulled from one and placed into another because the latter didn't pan out.
The main takeaway is that you have to be engaged in your kid's education; that's really the only thing homeschooling has, by definition, over other educational avenues.
Time for my anecdote. Our eldest daughter had a rough time with bullying at her public school a few years ago. She is, at heart, a nerdy kid who had issues wondering why other kids couldn't relate to her ... which is, for better or worse, like chum to a bullying shark in elementary school. We talked to the vice principal, her teacher, and the piece of dog ***t's parents. The bullying mostly came under control. And then, we tried to figure out what to do to make sure our kid didn't look like chum. For us, it involved two things she loves: dance and swimming. Dance gave her good posture and the ability to look confident and strong even when she's nervous. Lifesaving classes taught her to be in control of a situation. And now, she's a confident teenager whose former bully politely ask her for tutoring (and she gets taken out, gratis, for tea and pastries).
If we had pulled her from her public school and homeschooled her, I'm not sure she would've learned that degree of self confidence and poise. And she still would have looked like chum in a private school, but we would have been paying for her to be bullied by well-heeled brats.
FWIW, that's not what my parents did under similar conditions ... and I HATED elementary school. My parents were oblivious, and I've never really learned how to deal with difficult people.
Our other daughter is a different kid; she got an award from her school last year mostly for standing up for classmates who were being bullied. She's ... intense. Public school is a good place for her to be.