There are 2 major areas to focus on. First is what
you can do better to make the parrot
more motivated to come to you. But the second is about what you can
take away to make the parrot less motivated to go to the other places. I have written many articles about managing flight and hope you will take a look at them. What I'll post here is a supplement to the information I've already posted throughout that may be more specific to what I'd do in your situation:
1) Diet modification. You definitely hinted on some of this but this should be explored further. You want to make sure that more desired food is strictly fed outside the cage as reward for flight recall. However, you do want to continue offering healthy good in the cage to balance the "candy" you'll be the giver of. You want the bird to worship you as the giver of all things good. You may need to experiment with your bird to find out what it likes
now. Tastes changes and just because your bird loved something before doesn't mean it will be most motivated for it now. If you've been finding yourself using too much of what used to be a favorite food (like sunflower seeds) it may be time to take a break and find a new favorite food. I switch between giving seeds, nuts, or fruits as rewards to my parrots. They usually are more motivated when I switch to one they haven't had in a while but the motivation wears off a little if they get it too often. For an extra boost of motivation you can skip giving treats at all for a week. If you happen to be going away on vacation or have a week when you are too busy to train, don't give treats at all whether in the cage or out.
2) Food deprivation. What I described were all methods of food quality management, however, it may well not be effective enough to motivate your parrot for something as energy consuming as flight if its caloric intake is met in the cage. I hate having to sugar coat terms with BS just because if misread it can be misused. It can be misused even if it is read in detail so I will lay it out and it's the readers' job to use common sense and experience to apply it. Since both what I wrote in part 1 and now in part 2 both pertain to food management, I will refer to it as deprivation in this section. In reality it's only deprivation compared to what the bird is used to. Personally I see it as provision rather than deprivation. The amount I provide is limited rather than depriving cause my birds normally don't have more than the provisioned amount.
It is important to have a good control to judge motivation. If your bird knows any tricks it is very handy here. If it doesn't, I'd guess you'll have a much harder time teaching recall if you don't teahc some tricks first. We can judge training motivation by performance with basic tricks. I will use the
wave trick as my baseline standard since it is easy to teach and qualitative. If the parrot really really wants the treat, it will arise its foot sooner, higher, and for longer. If it doesn't want the treat at all it won't raise it at all on cue (given that it knows the trick and can normally perform reliably). Everything else is somewhere in between. If the parrot is doing anything less than a stellar job with wave which barely requires any effort at all, you can be certain it won't put in the effort of flying for that reward.
Using the trick/motivation method you begin reducing your parrots daily food. The safest and least intrusive method I think is unlimited scheduled feedings rather than actual reduction. The parrot would get to eat as much as before but with periods of time without food in between. If you can start with 3x 30 minute helpings of food, you basically can't go wrong. Test motivation prior to serving each meal with the trick method. If not effective, go to 2x feedings (morning/evening). Test motivation before meals. If still insufficient it is definitely a good time to make sure you're not doing something else wrong before proceeding with anything more substantial. However, if you are absolutely certain that it's overfeeding that causes lack of motivation rather than incorrect training technique or poor treats, you can try the following.
Don't leave water in the cage during meal time (but do leave it at all other times). Weigh the parrot immediately before serving food. Then weigh it immediately after it has finished eating to find out how much it actually ate. Weighing the food in the bowl is less effective because parrots are so messy that you can never be sure how much they actually ate by looking in the bowl. You can pretty safely assume the parrot can have 10% less than what it had been freely eating. Up to 20% is possible but if you have to go that far most likely other things are wrong rather than the amount of food. When reducing meal amounts you have to realize that amount of food in bowl never equals amount of food in parrot. So you can only reduce food served progressively by volume (or weight) until the parrot is up to 10% lighter than the base line weight you determined (and once again weighing for a week to find an average empty weight is better). So if you used to serve unlimited pellets, then you try serving 20 pellets and find 10 remaining, you can try going to 12 pellets, then you find no remaining but weight is the same. Then you try 10 pellets and find that the weight is 10% less, you know where to stop. You should be validating all of this with the motivation test method as well. Remember that the motivation test comes immediately before the next feeding rather than after the meal (because that's the best time to train).
So in the course of what I've written above I have talked about different methods of food management ranging from the least intrusive being saving treats for training, then scheduled feedings, and finally weight management being the most extreme. In none of these cases is the parrot actually being harmed or treated inhumanely because ultimately it gets the amount of food it needs, it is just that you have control over when it gets it and what it has to do prior to getting it. It is your responsibility to lower your training expectations to the point that it can earn 10-20% of its caloric needs from you. So if it means shorter flights, simpler tricks, or something as lame as targeting around, you have to give it chances to earn treats by doing something right. Practice and patience can always improve all this.
3) Reduce the distractions and "anywhere but to me" places. The absolute best place to train flight recall is a bare empty room with absolutely nothing in it so the parrot has no place to land but the floor. Sometimes a basement or hall way is suited for this. If you have no choice but doing it where you live, you can start by closing off everything it likes possible. Put those children protecting plates on outlets,
put possessions away into sealed drawers or boxes. I am well aware that it is hard if not impossible to secure everything but you definitely have to try your best. If you weren't so far away, I'd suggest
Parrot Training Perches but unless you're willing to overpay nearly $100 for shipping, you'll have to look into making something on your own. Definitely create more parrot designated places in your home. Invest in a ton of toys so that the parrot is having a good time on there (I know it still doesn't outweigh roaming the house but it's a start). Begin target and trick training the parrot on a designated perch for training. It should be distinct and distraction free.
Use the step from perch to perch and eventually spread distance to require flight method of teaching the basics of flighted target. Then expand this to recall. Refer to the articles on how to do this. DO NOT LURE, definitely use targeting instead. Luring just teaches the bird to go where the treats are. Targeting is an associative concept so if it can target some place it can also learn to associate your hand/cue to recall the same. But if you lure, when the treat isn't there, it's nothing to associate to so when it's gone, the behavior is extinguished.
4) Keep the recall command distinct from step up. I use the top of my hand pointed toward parrot perpendicular to the floor for step up. So I reverse my hand to show the bottom to recall instead. This is a very important distinction because you don't want the parrot to hop to your hand when reaching in because it learned to recall instead. If you pay close attention to the way I have my parrots
step up vs
recall in my videos you will understand what I mean as well as how I hold them. A Gray Parrot is similar in size to a Cape Parrot so this should be very helpful. However, a Gray's feet are bigger so unless you have big hands, you'll need to find a way to accommodate. I don't suggest using your arm though because it's less stable and when wearing short sleeves that scratches worse than being on hand.
5)
DO NOT PUNISH THE PARROT!! Anything you try to use aversive against your parrot during this stage will absolutely and definitely work against you. Any kind of punishment/aversive you have heard to be effective could possibly work for you once the parrot is very reliably recall trained. However, anything scary/bad/undesirable happening and the site of your face will work against you because the parrot will learn to fly away from you to avoid it being done to it. Whether its cage time outs, earthquake, squirting, etc., it will just reinforce that flying around so you can't get it will prevent those things from happening. You must exclusively rely on positive reinforcement and prevention (putting valuable things away) until you can have a very very bonded flight relationship established. Then punishment may work or may not but at least you won't set yourself up for guaranteed failure up front.
Going back to the cage must be a very positive experience or the parrot will always fly away from you when it is time to go back to cage. Follow the food management techniques above to guarantee that back to cage time after training is always desired. And overall try to make it that you are always doing things desirable to your parrots (even if it means faking the undesirable by leaving less food in the cage). You want to be positively associated and seen as the giver of all things good.
I think everything I wrote here equally applies to clipped parrots as well but owners don't care about their parrots well being so they don't go through all this trouble since they don't have to deal with it being able to get away from them when they want to force it to do something against its will. Positive reinforcement based training is to make things such that the things you want are wanted by the parrot because it is to its benefit and pleasure. Good luck and let us know how it goes. Please read all my flight related articles as well as
basic taming article because without them I may have left many gaps in this reply and it would not be sufficient for proper flight training.