I'm partial to amazons, hot 3 amazons in particular (I know, many people avoid them, but I like them!). I appreciate their more 'independent' nature, the fact they are less likely to develop behavioral issues like plucking/screaming and the fact that they show very clear signs when they are feeling aggressive so you can (for the most part) avoid being bitten. I'm not too into having a 'lap bird' and chose the tendency to self entertain over affection in a companion parrot. I grew up with a goffin and can honestly say, I wouldn't consider a cockatoo personally. Alfie is very loving and cuddly towards my dad and even has his moments with other people where he wants to be petted or play. However, he is a 'stealth biter' giving little or no warning when he's feeling aggressive. When he gets you, he won't let go until you physically pry him off. I'm not sure if it's the fact he doesn't let go or if his beak is actually sharper than an amazons, but he inflicts the worst kind of bite (often to the bone). Zons seem to strike and let go. Alfie also chews anything he can get his beak on, and from my understanding, that's pretty normal for any cockatoo. All the amazons I've known weren't excessively destructive, seemed to pick up on the fact they could chew up toys but not rip up the house and have been able to spend most of their days out of their cages so long as someone is around and not cooking/opening doors. Alfie has never been able to be out unsupervised for even a second. And this is strictly a personal thing because both species are extremely loud, but I prefer the sound of an amazon on full volume to the sound of a cockatoo on full volume. While probably comparable in decibels, there is a difference in pitch, and which one you find most tolerable should be a consideration as all large parrots will vocalize loudly at times.
I have not spent significant time around any grays to have a real opinion. They were in my top 3 species when looking for a bird though. One thing to keep in mind is their intelligence. I struggle sometimes to keep Kiwi busy with his level of intelligence, and he's no where near as 'smart' as the average CAG. It's apparently not that uncommon for well loved CAGs to pick up a large enough vocabulary to be able to speak in context to communicate and learn concepts on par with what a young child can. Puzzle toys and learning new things may be easy the first year or 2, but it can be difficult to entertain a creature perpetually trapped in the mind of a 4 year old for the next 60 years! Then again, how nice would it be for your bird to possibly someday be able to tell you what it wants instead of just scream until you guess right? Just something to consider