Cockteils are great birds. They can be hand tamed relatively easily, but it will be harder in a big cage, as it will have less exposure to in. In a small cage in the house, it will get more used to people walking around it all the time. It is great for them to have room to get out and stretch their wings, but maybe it would be better to do this when it is a bit tamer? Otherwise to catch the bird up again in the avairy will probably require a net, and this could be very stressful for the bird, and possibly discourage it from getting close to people.
Cockateils are definetely better off on their own, or with other, larger parrtos. Anything small that you put in with them, they may tend to kill. We bred ringnecks, budgies, canaries, finches and quail, all together in a huge avairy, at various levels. We had cockateils in there too, but had to sell them, as they killed babies, kicked eggs out of nests, and bullied the other birds.
They make wonderful hand-tame pets though, but sometimes become very attached to one particular person. Has yours been hand-reared? It's often easier to tame them if it starts from birth.