I use a base diet of Harrison's Adult Lifetime and supplement with a variety of fresh vegetables, mostly those I'm eating myself. (Obviously you never, ever want to feed avocado, chocolate or most fruit seeds, and be very careful with non-organic produce, as birds are more immediately susceptible to pesticides than we are.) I also offer the occasional piece of cooked chicken, or live mealworm or cricket. Wild parrots do eat some grains and a limited amount of seed (Australian ground-feeders more than most -- I'd limit your GCC to the occasional seed).
I don't feel that we know enough about wild parrot diets, especially species-specific diets, to feel comfortable feeding my birds a homemade diet. The flock I know best has been observed eating cherry blossoms, strawberry guavas, juniper berries and all sorts of things I wouldn't know how to get -- and that's certainly not the full extent of their diet. I do feed my cats a raw diet, but cats are easily observable in the "wild", have a fairly limited diet and show illness easily, whereas parrots eat a huge variety of foods, are prey animals and have evolved to hide illness. You can barely even tell when they get fat, unless you are actually weighing them every single day.
Here's a fun anecdote -- make of this what you will:
We had four baby cherryheads from the wild flock, still in gray fluff, turned into San Francisco Animal Care & Control -- they were probably pulled out of a nest somewhere, we don't know. They were taken in by a member of a local rescue to foster and raise. Since the mother lays all her eggs over a short period and not all at once, the birds were different ages and matured at the same rate, consistently taking milestones in what we thought was hatching-order. The youngest bird gained the most weight and built the most muscle, and to this day is the most robust and healthiest of the birds... and incidentally, that bird was the one who was started on juvenile formula earliest. The one we think is oldest turned out to be the smallest bird.
Unnatural? Probably. Beneficial? I think so. I can't ignore the correlation between size/strength and length of time on juvenile formula. I don't know what the makeup of food from parents looks like, but I'm guessing it's lower in protein and vitamins than the formula.
If you must switch to a non-pellet, homemade diet, please don't go it alone -- talk to your avian vet (not regular dog & cat vet) at length about proper diet makeup, and look on the back of your pellet package to figure out appropriate vitamin, fat and protein ratios.