First of all, welcome! This is a great forum with so many opinions and lots of different experiences etc---you will fit in perfectly! Glad you found it!
Cockatoos are "needy" because they are so social and can be so sexual (so you have to have a LOT of time for them--and interaction that does NOT revolve mostly around petting/sitting--which can easily come off as mating behavior). They are like larger, MUCH more intense cockatiels (same family) with significantly greater noise/damage capacity and amplified behaviors. Even though they want cuddles more than other birds, YOU SHOULDN'T CUDDLE THEM (as it is very sexual and leads to the behavior problems that lead to them being relinquished by owners). Although the Galah isn't at the top of the list when compared to the larger ones, as a collective, cockatoos are re-homed more than any other bird species (world-wide and nationally within the US).
You indicated that your tiel became hormonal---a cockatoo's hormones are a much bigger problem in terms of noise/danger/craziness (and they are even easier to stimulate hormonally) etc.
No bird should be cuddled, but especially not "toos". Head and neck petting only (cockatoos are known as "cuddlers" but they get crazy about them very quickly--in an unhealthy/sexual/aggressive way). No shadowy spaces, no wrapping in towels, no drawers, no under the covers/pillows ..
A conure is probably a safer bet given your desires, but still know that cuddles are no good for them and can lead to sexual/hormonal behavior at puberty. They still shouldn't be cuddled---dogs/cats can handle it, birds cannot (not past puberty).
If you are the type who
wants to cuddle, my instinct says, don't get a cockatoo, because it will be very hard for you to draw that firm line that needs to be drawn (i.e., no cuddles from a cuddle monster). They are so cute and it seems innocent (until it isn't, and usually, by then it is a problem). That does not mean that you can cuddle a conure either, but there is a bit more lee-way.
In terms of housing etc---being the owner of a galah will mean that you will need a huge cage/aviary and a ton of time/energy..for a single galah, you will need to be like a cracked out pre-school teacher planning enrichment activities all day for 40+ years...
You already indicated that your last bird pulled away when it got a mate (which often happens) and so if you plan to be this bird's everything (main companion), then you will need to be VERY ready to prep lots of creative activities (and provide very frequent socialization opportunities) because they can be driven by motors and while they are hilarious and fun, they aren't exactly relaxing most of the time (i.e. cuddles).
Don't get an unweaned baby of any sort. It makes no difference in the long run whether you or someone else feeds a bird so young....Read the thread suggested above. Yes---a bird must be hand-fed by a person for the easiest training later on, but it doesn't matter who does it. All babies are easy-going....SO MUCH can go wrong and it is crazy expensive (for any parrot). Also, hand-raising quail etc isn't going to be the same...Even though it is cool that you have that experience.
Especially since you indicated that you are considering buying a baby and you already want a cuddle-bug, I would steer you away from the cockatoo (again) because they hit puberty MUCH later than most birds (which leads owners into a false sense of comfort regarding their own -often inadvertently harmful-behavior with the bird) and if you set a precedent when they are young and make mistakes with these needy and sweet baby balls of fluff, they will be a nightmare within 5 to 6 years (and you will be so confused/unhappy with the changes unless you set firm boundaries and teach extreme independence early on). They really don't understand that you have a life outside of them and want to attach to your hip UNLESS they are taught to do otherwise and given plenty of time to fly/run/hang/flap etc--- in addition to interaction that doesn't spell sex to them...which is way easier said than done when they are very persistent.
I'm moving with my dad in about four years, and yes it will be permanent (my grandparents have been citizens for six years and they live in Colorado). Dad's paying off the house. I may leave my cockatiels in the care of my sister, as I'm not sure that they would survive the stress of air travel. I was in a really bad place when I first got them (mental health wise) so I didn't take the time to socialise them or get them used to new situations, and consequently they are very easily stressed.
Please know that if I get a new bird I will not make that mistake, and I'm doing way better then I was two years ago.
^MOVING IS A MASSIVE STRESS to even the most perfectly socialized birds and if you know it is happening in 4 years, PLEASE wait. The fact that you may leave your tiels is a bit heartbreaking for them (and probably hard for you) and I want you to know that even if they had been more social, travel is not easy for birds..So is being left behind (I am not trying to make you feel bad, but you seem confused about how this works). A very tame bird can have a heart-attack in a plane (not that they all do, but it IS a big deal for them. A bonded bird leaving its main caretaker also leaves a permanent scar...So, think about that before you dive back in.
You also said, "when I move to the US" ---any moving with birds is VERY stressful and tricky (especially internationally)---not good for them unless you have no choice. The thing is---a lot of people have to re-home their bonded birds when relocating due to the cost and craziness of paperwork and logistics (you can be denied by the country of entry or exit and breaking a bond to leave your bird behind damages them--even if they go to a new home). If you are even planning on moving internationally within the next 5-10 years, I would strongly suggest that you WAIT to get any sort of parrot until you are more settled (for your sake and for theirs).
Oh--and I wouldn't assume any bird can stand ground with a cat (ever)--I know there are lots of videos of cockatoos with cats, but their claws and saliva contain bacteria/parasites that are deadly to birds (a paw in the bird's water could kill it, as a cat paws its litter and those parasite/bacteria get onto their feet/nails). Many birds die from infection after a minor (even playful) run-in with a cat --see
https://nilesanimalhospital.com/2013/01/22/2503/ (Pasteurella multocida-among others). Plus, a galah (while bigger than a parakeet) is not that big overall...I mean, yes, they are larger, but it's not like we are talking about a macaw...still way smaller than a cat.