Welcome to the forum, Jake and GCC! You are correct, telling you to be patient without giving you any instructions doesn't help you or the bird so let's see if we can help your bird a bit but, first of all, let me tell you that a month is NOTHING when it comes to parrot timetables. It's a lot to us, humans, but nothing to them.
Liz is right, you have a baby that has been taken away from everything it knows and which made it feel safe so it falls to you to convince it that your home is a good one and that you are the bringer of good things and not a giant alien who wants to hurt it. Unfortunately, you have made things worse for the bird and yourself by putting your hand in its cage. This is called a flooding technique and they are now no-nos in the parrot world. A flooding technique is any method of taming or training that imposes our will on them without giving them the chance to choose. When you put your hand in its cage, the bird has no choice but to accept it for the simple reason that it cannot get away from you - it's your way or the highway and it doesn't work with parrots in the long term because they don't understand the concept of obedience, submission or discipline. Dogs do and although I would not recommend a flooding technique with them, either, it doesn't hurt the relationship the way it does with a parrot. You see, parrots don't live in hierarchical social groups where there is a leader or alpha, they are all 100% equal and no parrot imposes any rules or behaviors on any other parrot so, when you try to do it, all you are doing is confusing/upsetting the parrot and convincing it that you are not good. Also, gloves are not good. They are now considered so bad that even people with parrots that bite hard are discouraged from using them. Parrots seem to have a difficulty accepting hands so anything that disguises the fact that these things hanging at the end of our arms are part of us or makes them scarier are not recommended.
This is what I would recommend: open the cage door at dawn and let the parrot come out on its own. If you have a door that opens to the side, put a perch outside the cage that is easily reached from the door itself. If you have a door like a 'moat bridge' (the kind that opens down and locks in a position parallel to the floor), you don't need to but I would still put branches tied to the side of the cage going up so the bird can climb and go high (there is safety in height). Is the bird clipped or flighted? Because a clipped bird is an insecure bird and insecure birds take longer in trusting so I hope your bird is flighted both for its and your sake. Offer it a nice piece of fruit (GCCs are mainly fruit eaters in the wild so they need to consume large portions of them every day and should never be free-fed protein food) but, if it doesn't take it, just put it somewhere where the bird can reach it easily and just walk away - same thing with any treats you offer during the day. These offerings are not rewards for anything done well, they are tokens of friendship and goodwill and should be given just because. Spend as much time as you can in the same room that the bird is in - and, I am warning you that, for GCCs that means 4 to 5 hours a day. GCCs are one of the neediest parrot species there are so they need an inordinate amount of time spend with them or they become screamers and biters. Talk, sing. whistle, offer it a treat and just keep it company. Don't ask it to step up or anything at all and don't stare at it, always check to see what it's doing out of the corner of your eye because staring is something only predators do. In time and as the bird learns to trust you implicitly, you will be able to look at it straight but not for now because it doesn't trust you at all so you want to reinforce the fact that you are not going to hurt it and give it time to make up its mind about you.
I assume that you are offering soft food every day so what you can do is do it from a spoon (it's a great bonding technique) but you will have to wait until it starts taking treats from your hand without any hesitation before you start.
Now, just reinforcing something that I mentioned at the very beginning of this post: give it time. When people said you need to be patient, they meant months and not weeks. It takes a parrot about two years to feel 100% comfortable and settled into a daily routine (and this only if the daily routine is always the same!) so, as you can see, patience is essential with them so, please, tell your wife not to give up, that the game hasn't even begun yet!