I am afraid that you got yourself a bird from a bad breeder. Not that any breeder of parrots is good, mind you, because commercializing a living being with feelings is not what one would call a 'good deed' or a favor to the animals but there are degrees of 'badness' and not hand-feeding is a sure sign of not good at all because it shows a complete disregard for the animal's wellbeing in captivity. See, the thing is that taming never achieves anything closer to what imprinting does and nothing you can do or not do is going to change that. You can tame a parent-raised bird but the bird will never develop the kind of bond with humans it will need to be semi-content in captivity - the ONLY option for happiness in an IRN that was parent-raised is for it to be an aviary bird that lives with another of its own species. I am sorry to burst your balloon this way but this is not my personal opinion, it's a scientific fact (do research on sexual and filial imprinting on highly altricial social species and you will see).
Now, this does not mean that the bird cannot learn to relax in your presence or that it will never trust you at all! You can get that but it will take a loooong time, a lot of work, a lot of patience and, in the meantime, the bird will suffer because it won't have anybody or anybirdy. Again, I am sorry to be saying these things to you because it's obvious to me by what you posted that you do want to give the bird a good home and that you were not aware of the huge difference between a parent-raised and a handfed parrot so you never really consider having a bird living and suffering all its life in your home but highly social species need not only constant company and interaction with others and a lone parent-raised bird does not have this. Again, not my opinion, it's a scientific fact that highly social animals have certain needs that territorial animals do not. You can keep a very happy canary all of its own but there are no happy parent-raised parrots all of their own. The only way a lone parrot is content in captivity is when it's a companion species that was hand-fed and has been cared by a loving human with excellent husbandry.
As to the sitting for '15 minutes without looking at the parrot'... well, in all honesty, I do not know how that improves your chances because sitting next to a scared bird in a cage is a form of flooding (after all, the bird cannot avoid being closed to you so you are, in truth, forcing the bird to accept your company and not giving it any choice at all). I do not do that. I actually do not do anything when I get a new parrot that doesn't trust humans. I simply allow time and consistency to do their 'magic'. Parrots are highly intelligent and they think the same way that humans do -meaning, they put two and two together and get a four out of it. If you want to do it, I would think you will have to wait until the bird is no longer afraid of you so when you see that the bird does not move away from you (or your hand) when you are close, when it starts perching on your head when it's out flying, you can start sitting next to it when it's in its cage... but, again, I do not see how that would improve anything because, by then, the bird is no longer afraid of you. I'll tell you what I do (and not only with parent-raised birds but also with birds that were abused to the point that they fear and hate all humans). I leave the bird in its cage for 2 weeks or so without paying any attention to it. I clean the cage, put fresh food and water and simply walk away and do my own thing. During those two weeks, the bird starts to relax because it sees that day after day after day, I do not even look at it (I do look at them but they don't see me doing it), it gets used to the right diet, I figure out what its high value item is and it gets used to the same exact routine every day: windows uncovered when there is the smallest amount of light in the sky (with NO artificial lights on, of course), cage doors open for them to come out to fly (they never do come out immediately because there is not enough light for them to fly but even when there is light, some of them wait a minute of two before they leave the cage) and, once there is a bit of light, I start cleaning the cages -again, no artificial lights on. I take out the bowls and remains of the fresh produce they got the day before, change the paper at the bottom, scrape the poop and food off the perches, etc), then I put out fresh water followed by fresh produce and finally, the gloop. Immediately after putting out the gloop, I turn on the ceiling lights and start putting them back in their cages so they can eat their breakfast. They stay in for about 15 or 20 minutes and then they come out again and stay out until they go back inside at 2:30 pm this time of the year (I close their cages so they cannot go back inside and munch on gloop -there is raw produce easily available for them to eat, just not gloop- because this way, they are a bit hungry and go back willingly). They then get dinner at 3:30 pm (this time of the year, the days will start getting longer after Dec/21 when we have the winter solstice and the time of their breakfast and dinner will change with the seasons). The trick is to teach a parent-raised bird which does not step up to a hand or a stick to go back into its cage when told to but, again, consistency, persistence and -most of all- patience always wins - especially when bribed with their high value item
.
It is easier for me because the parent-raised birds see the handfed ones interacting with me without any fear whatsoever and, because they have companions of their own (the quakers have each other and the GCC loves Epuish, a male white belly caique with whom she flies around, cuddles up to, allopreens, etc), they are not lonely. But the other reason why it is easy for me is because I have no expectations whatsoever in terms of making them my 'pets' so I don't put any effort whatsoever in 'taming' them... it just happens. I don't have 'pets', I have companion animals that live with me and which I take care of. I am just here to give them as good a life as I possibly can and not because I 'want' a parrot - which is not the norm when it comes to parrot ownership. I do the same with my dogs and cats - if they need a home and I have room, I take them. And so I have dogs that require diapers because they never did learn to go outside and one that requires a muzzle when he goes to the vet because he never did completely 'shed' his aggression for strangers, cats that don't like other cats (because they were not socialized properly when kittens), cats that don't like people (because they were feral cats that lived in the street and were never socialized to people) and parrots that cannot be handled. I've been doing this for many, many years and have learned that love and time have a way of fixing things if you don't rush things. There is a saying in Spanish: "Dale tiempo al tiempo" (give time to time) And it works every time.
As to the cage you got... well, it's a passerine flight cage (for canaries or finches), that's why it doesn't have the little trap doors. I have flight cages too but mine have little doors in the front and the sides.