As someone who was homeschooled off and on (and finished the last ~5-6 years being homeschooled and working part-time), I agree that the answer is "it depends" - this can't be overstated. It works for some kids and not for others, and probably some parents and not for others. I was perfectly happy to have my nose buried in books most of the time and I liked learning new stuff so it worked pretty well for me (and the part-time job in my parents' store gave me plenty of interaction with people of all ages, ethnicities, even languages). My brother, on the other hand, hated doing schoolwork and my parents struggled with him a lot just to get him to do the basics and I think he would have been better off in a more structured environment, realistically - plus, I think he really craved much more interaction with kids his own age than I did, so it was a lot harder on him to be at home and/or primarily around adults all the time.
So agreeing with the parent, the summary question is really strongly dependent on the actual people and kids involved. It probably wouldn't hurt to try out both options. Ultimately, whatever is making the kid happiest and encouraging him/her to learn the most is what's ideal, and I'm not certain that you can make that judgement without having tried both courses.