Marie, when people talk about an animal been more receptive to training because they are hungry and the reward is getting food, you are talking about a HUGE lack of empathy and a HUGE control/power trip. Have you ever heard of dogs or horses having to wait for their dinner so they can be more receptive to training? No, you haven't. Dogs and horses are trained in the middle of the day, after their breakfast and well before their dinner (if they eat twice a day). It's actually counter-productive to train an animal when it's hungry and the only motivation for performing is the food they will get for it. Animals, like people, cannot learn well when they are hungry.
The reason why people use it with birds (see note below) is because birds don't feel the need to please anybody, humans or their 'leaders', so the only way of ensuring a consistent response is to force it - thus, hunger been a great motivator. Otherwise, given a parrot a choice (and that means self-confidence, flight, outside a cage, etc), he/she will only do what you ask when he/she feels like it. Humans tend to be very controlling and want all their pets to obey them all the time perfectly regardless of the situation, time of the day, animal's own volition or mood and whether they (the humans) deserve the obedience or not. And parrots not been hard-wired for obedience present a problem for the human who needs the immediate gratification of show of power over the 'inferior' animal. Thus, the food management technique.
Note: people used to do it (and I am sure they still do) with canaries, they would not feed them for an entire day or more and when they were starving, they would put their hands in the cages with food on the palm so the bird, desperate for food, would perch on the hand. They learned but it's a despicable thing way of getting an animal to do anything!
Now, let's not confuse weight management with food management because bird people are now using them indistinctly, but they are two completely different things. Weight management means exactly that: managing weight. It consists of switching food around (like low carb items instead of high carb), not free-feeding, reducing portions and increasing frequency in order to keep a faster metabolism, feeding fiber items when hungriest and carb and protein later, allowing for exercise, etc. It's done solely for the bird's benefit and I practice it all the time and have been doing it for many years. Food management, on the other hand, is nothing but a euphemism for 'get the animal to perform no matter what and, if you have to, make it hungry", it's done by reducing portions without making up for the quantity taken away, eliminating or delaying a meal, etc. and it's done solely for the human's benefit. One is good, the other one is not. And birds that learn under food management can end up with problems; the most common one been that the bird reverts to baby begging behavior almost all the time he/she is with a human -which might sound like a very insignificant thing to put up with but, in reality, it's not because it means the bird is under severe distress.
I am all for training birds. I do teach my birds certain commands and they all learn whatever I want them to but I do it the same way I do it with my dogs, cats, children, grandchildren -and husbands

The commands are all necessary for cohabiting (or, in the case of humans, simple good manners or consideration for others) and the teaching is done when the occasion arises and through repetition, consistence, persistence and positive reinforcement. None of my animals gets a treat for doing something right (well, the wild-caught might at the beginning but then I wean them out of it), they do it because they want to. And the treats or food are NEVER directly related to performance. Food is their right and providing it is my obligation and not to be used as a way of getting something I want out of them.