Galahs are certainly quieter than most cockies, however they will flock call in the mornings and evenings. I'm sure you've heard their shriek when wild flocks fly over. That shriek can indeed make your ears bleed, but as I said: it's only for a while each morning and evening. Sometimes, they'll yell at wild birds if they call out, but mostly they're good.
My poor old bloke rarely called and when he did we all came running with treats for him. My experience with Dominic makes me certain that galahs are the nicest of the cockatoos to keep. SC2s are very, very needy and not great at self-occupation. My corella is easy, but she's very active and also happens to be the noisiest bird I've ever heard. Her flock call makes my teeth want to jump out of my head! Honestly!
The black cockatoos are said to make good pets, but the only ones I've known to become truly tame have belonged to people who totally concentrated on them for long periods. Also, they're kind of out of my financial range.
Major Mitchells are gorgeous and I've known a few really lovely pet ones, but you need a licence and they're every bit as noisy as the other cockies, calling over long distances.
You might consider Alexandrines? I have a pair of those and they're lovely: very confiding and trusting and affectionate (although they don't much like being petted). The hen, Madge, flock calls twice a day. It's a piercing sound, but not as ear-bleeding as the cockatoos. Eclectus are lovely but they do fruit-poos, so that might put you off. Or Rainbow Lorikeets? (Also fruit-poos, though).
Really, it's up to you. If the noise factor is the deal-breaker, I'd probably go with a galah. You'd need to know, though, that you might end up with a vocal one and that might just not suit your son.
Why don't you take him to a wildlife sanctuary so he can listen to the various calls? Try to get there as early as you can, or linger till dusk so you get to hear the flock calls. I know how much this matters, as I've experienced ASD-overload meltdowns with my son. The thing is, yours can probably become acclimatised over time, but it has to be in *his* time, not the bird's.
You don't say how verbal/communicative he is, but perhaps you could negotiate with him to try a few things (eg. visits to breeders or sanctuaries)? I know my own son is often reluctant to experience fairly mundane things and has the capacity to build up irrational fears before the moment. That can usually be defused with some calm conversation, however your situation is obviously going to be different. It's often just a question of exposure over time (although you know that already).
Might it be at all possible that your bird could live outdoors and come in for periods during the day? That's what I do. My birds live in cages on the enclosed deck and I can bring them indoors if I wish. Oh Lord! I can't imagine living with Rosetta bellowing inside! We'd all go mad! LOLOL! I thought Madge was a bit OTT, but Rosetta has brought a new level of skraark to our house.
So, have a think about it. There's no hurry but remember: if you acquire a cockatoo you'll have him for your lifetime and probably beyond.