You are absolutely right about the reason why dogs [or cats] should never be free-fed but I am afraid that you are missing the reason why parrots should not so let me explain my reasoning to you and you decide if I am right or not. The problem is not whether the parrot will eat all day long but what the parrot will choose to fill its crop with.
There are all different species of parrots and all have different diets, some eat more grass seeds [like cockatiels and budgies, for example], some eat more nuts [like macaws], some eat only nectar and pollen [like lories], some eat more fruits than anything else [like most conures], etc. GCCs happen to eat mostly fruit which means VERY low protein and fat and VERY high moisture and fibre. They do eat seeds but the seeds they eat are what we call 'green seeds' which are the seeds inside the fruit they are eating or grass seeds which are low in protein and fat. Parrots, like any other animal, need protein for breeding [egg yolk is very high protein and this is what the baby bird uses to grow inside the egg] as well as for life [muscles, feathers, energy, etc] but protein for herbivores is not found easily or in abundance in nature. Birds don't eat raw beans [they are toxic] and even if they do find an entire field with a crop of sunflowers, the crop is available only once a year so nature gave them a craving for it and 'programmed' them to eat and eat and eat whatever source of protein they find until there is no more. THIS is the problem with free-feeding them high protein food!
I have parrots and I also have passerines that are natural seed-eaters [canaries, finches and a cardinal] and the difference in what they crave and choose to eat first and most of is clear. Feed the same thing to a parrot and to a canary, say, a bowl of seeds, a leaf of romaine lettuce and a piece of apple, and the canary will ALWAYS go for the lettuce first while the parrot will always go for the seeds! I give my canaries a leafy green portion that is twice as big as they are and they will eat the whole thing by the pm but I can only get the cockatoo and the little ones [budgies, plets] to eat a decent portion of the greens without trying. See what I mean? It's not that GCCs will not eat fruits and veggies if you free-feed them seeds or pellets, it's that they will end up eating way too little of them and way too much of the protein food.
I do free-feed my parrots but not high protein food. I give them large portions of gloop and produce for breakfast and all day picking and, for dinner, I give them a measured portion of their seed/nut mix -enough for them to fill their crop well and a tiny bit extra, just in case they are hungrier that day. This way, I ensure they eat low protein, low fat, high moisture and high fibre while still providing enough protein.
Now, as to a GCC being a picky eater... well, it's all a matter of timing, preparation, observation and persistence. I only have one now, a female named Codee, but I had four of them and although all came to me as seed junkies, they all transitioned easily to a healthier diet -and very fast, too! In my personal opinion and experience, GCCs are excellent eaters! They are not on a level with cockatoos or amazons but they are pretty much up on my 'good eaters' list. Timing because they are hungriest at dawn so this is the time when you give them the healthy food. Preparation because they are all individuals and some like a chunk stuck between two bars, some like things cut smaller and skewed, some like things chopped or sliced and put in a bowl, etc. Observation so you can figure out what the bird likes best and persistence because, to a parrot, a new food is not familiar until it has seen it 1,000 times! And I firmly believe that persistence is the problem with people who say their bird will not eat something. People tend to try giving them, say blueberries, once, twice, three times and even ten times but then they stop convinced the bird will never eat them - like when you say 'It's not going to work' but it took me five years to convince an African grey to try blueberries even though she got them once a week throughout the entire five years and all birds love them to pieces. Now, I will admit it's easier for me because I have a number of birds so the new ones see the older ones eating and they eventually will try the new item [parrot see, parrot does] but eating their food in front of them does the trick, too!
As to BeBe (I don't know if you pronounce it Beebee but, if you do, that's the name I am called by my family
] not liking wet or mushy things, I believe you! My birds don't like mushy or wet, either! But gloop is not mushy or wet. It's a dish made out of cooked grains but the grains are cooked al dente so they retain their shape and integrity, remaining separate one from another (think of parboiled rice] and I drain them very thoroughly as well as the stuff I use from cans which I also rinse under running water very well. What I am trying to say is that, if I put a ladleful of gloop in a colander, no water will come out of it -not a single drop!- BUT all the ingredients are infused with water so as to make the dish high in moisture so as to emulate the same water content of the diet they would get in the wild - see what I mean?
As to getting 'fancy' stuff online... I no longer order much online because even my regular supermarket [Shoprite] has most of the stuff I need for their diet (they have a 'healthy aisle' with organic stuff and such] and I round it up with stuff from Whole Foods [like the black lentils or frozen organic blue curly kale which I cannot get in Shoprite but you can make it with regular lentils and no kale, too]. The only things I order online are things I cannot find easily like naturally dried, no sulfites, organic apple chips, for example.