It all depends on the child.
I agree entirely about the investment. But if your dc are unhappy and you can't move them elsewhere or somewhere suitable, then it might be worth it, or if they have a special talent, sport music academic, whatever.
Tbh, if they are motivated and supported, they should do well in most circs unless something awful happens.
We do send our Ds to private school, for a number of reasons. But one of them is that he is dyslexic and didn't get into the only grammar school in the area, and the local school was pretty dire. It was then a question of spending about 400k to get into the catchment of the next reasonable school, actually a v good one, or working out for what appears to be short term benefit but is doing him well. It ends up being baked beans for supper either way.
His class is prob more diverse than the local one would be (masses of bursaries) and the facilities are good. The teachers vary, tbh and in any case you get on with some and not others.
He does go to a pretty selective school. Again, tbh, I would think twice about sending a child to a private school just because it is private. There are quite a few that are worse than nothing special. And don't assume that just because the facilities are there your dc will use them, they may or may not. If your dc shows a talent for something or has issues like ours, then fine.
But tbh, if he had got in to the grammar I would have been delighted, banked the money and spent it as you suggest or supporting him on other things, eg on trips abroad to learn French etc.
It's certainly not worth putting yourselves under stress for if you have other reasonable options.
And I was privately educated....
Sorry this is long.