I've been selling houses for 28 years and I've seen it all, I've even camped out with my kids while they were learning English vocabulary 😄 but look, the thing about private lessons at 9 years old really depends on what your kid is actually like. if he's the type who enjoys learning and asks questions, one class a week can be useful to reinforce what he's getting at school and give him a bit more confidence. now, if he's just getting by "okay" but without frustration, he's probably fine where he is.
what I would do is pay attention to whether the kid shows curiosity or if he struggles with pronunciation, grammar, like the basics. if you see he avoids speaking in class or that he's falling behind, then yeah it makes sense to add an extra class with someone who comes to the house. but if it's just because other parents are doing it... that's kind of like when everyone wants the most expensive house in the neighborhood without actually needing it 😅 plus at that age, 30-40 minutes of good extra exposure is worth more than two hours of boring videos.
my advice: try something cheaper first, like an app or youtube videos focused on his level, and if you see he gets bored or that he really needs it, then you find someone. sometimes kids make progress on their own in july and august without classes, which is weird but it happens. give it one more term at school and you'll already know if it's necessary.