OK. First of all, you are expecting too much too soon. Three weeks is NOTHING to a rehomed bird. As a matter of fact, I never even try to interact at all for the first two or three weeks. If you are a dog person, you need to understand that parrots are nothing ike dogs (I have dogs, cats and birds and although I now consider myself a 'bird person', I used to think that I was a 'dog person'). You can take in a dog that belong to somebody else (all my animals are rescues, adoptions or rehomes) and, as long as he was not abused in any way in his previous home and you are kind, the dog will bond with you and love you in a matter of days. Not a parrot! For one thing, parrots are undomesticated and, for another, they don't understand the concept of subservience (or obedience or discipline - they don't belong, like dogs and humans do, to a hierarchical society) so, to him, you are a complete stranger and somebody he doesn't know if he should trust. Asking him to step up to your hand is like asking a complete stranger to trust you enough for you to grab his hand and take him wherever you want - and I bet you $1,000 against a penny that the stranger will not do it, either.
Your light schedule is good. Not great because it's not completely consistent which is best but not too bad. Just make sure that the is exposed to, at the very least, 1.5 hours of twilight both at dawn and dusk and that no light comes in through a window and reaches his cage in the night (moon and stars light is good, it's the street lights and the cars driving by that are not good).
Now, the diet is terrible. I don't mean to make you feel bad but the diet that you have him under will kill him way before his time. Let me explain. IRNs are fruit eaters so they need a lot of produce for the high moisture and fiber, VERY little protein and almost no fat at all. You are feeding eggs (animal protein which is not the same as plant protein!) and that means a bomb of protein, fat and, worst of all, bad cholesterol, something that their bodies cannot get rid of because Nature did not evolve them to ever consume any food with bad cholesterol so even when you feed a little bit and not all the time, it's still super bad because that little bit will stay there and all the little bits will accumulate and accumulate as time goes by, You are also free-feeding seeds and that's way too much protein and fat for him (and, by the way, avicakes and nutriberries are made out of seeds so they are no good either). You need to change his diet so I will tell you what I feed mine and you can then determine what you want to feed yours because there are other options (chop, mash, kitchen sink, etc). Mine eat gloop and raw produce for breakfast (they get their produce first thing and their gloop about 1/2 to 3/4 of an hour later) and get one level measuring tablespoon of finch seed mix for dinner (finch mix has the lowest protein and fat of all the seed mixes).
I have two females that came from a lady who had them for 6 years and who had taken them from still another lady - they have no rings on their legs so I don't really know how old they are but I assume they are around 10 years old because most birds are rehomed before they turn five years old (it's pretty much when the novelty wore off and the people realized they are too much work and trouble). And they had been eating the wrong diet all this time so I now have them on liver and kidney cleansers but is going to take about a year or more of daily supplements and good diet to begin to correct the problem (not that it will ever disappear, mind you, but the liver will get a better function). So I would urge you to change his diet immediately (I can tell you how to do it, if you wish). It's not really that hard with them because they do love their fruits with a passion - and mine absolutely adore their gloop (I have an albino and she always ends up with her entire forehead and cheeks dirty from her buring her had in the gloop
).
Now, I don't really interact that much with them still even though they have been here for several weeks because they were EXTREMELY high strung when they first came... I am talking plucking, night and day frights all the time, anxious and nervous all the time, etc. They were obviously not allowed to come out of their cage because they did not come out for weeks even though the door to their cage was open all day long and, when they did and they were startled into flying, they were super weak and clumsy. They have gotten better. They now come out every day, they are more relaxed in my presence (preening, eating, etc), are getting much better at flying and do it more regularly, step up to my hand and even allow me to give them head scritches (not for very long but it's still a step in the right direction).
I suggest you take it easy and don't rush things because this is what we call 'the honeymoon period' when the bird is at his best behavior (in terms of aggression) and not only getting used to its new home, human, etc but also making up its mind about you. It's the foundation where your entire relationship with him will be built and you need to make it a strong one, one made out of trust. So don't rush him (they don't like that) and don't ask him for familiarities (step up) that he is not willing to give you yet. Wait him out - show him the respect you would another human being and let him set the relationship at his own pace. It will work out much better for you and him in the long run. For now, just open his cage and let him out on its own, talk/sing/whistle to him and, every now and then, give him a little treat (make it a small piece of a millet spray) but, if he doesn't take it from your hand, just leave it where he can reach it and walk away because this is not a reward for good behavior or a bribe, it's a gift you are giving him, a token of your desire to become his friend.