Thanks for the video, it's helpful to know what kind of noises he's making.
I think the first step to make is to get him onto eating pellets. I know you've tried before, but we just have to keep trying until he gets the idea. I think it will make a big difference to his continuous calling.
What brands/types of pellets have you tried?
Where I used to work we fed this diet:
1/4 cup of boiled lentils, barley, beans, peas
1/4 cup mixed seed, sprouted and unsprouted (safflower, sunflower, corn, barley)
1 thick slice corn on the cob
1/2 cup diced vegetables (broccoli, carrot, kale, green peas, green beans, etc)
1/4 cup diced fruit (Apple, pear, grapes, papaya, etc)
1/2 cup Zupreem NutBlend
Nuts in the shell as treats
Zupreem pellets are hard to find in Australia, Vetafarm's maintenance diet is a fair substitute, or if you can get your hands on Harrison's Adult lifetime or Pretty Bird Daily Select they would be equally good.
Now, I'm going to list a couple of different 'methods' of switching an underweight parrot onto pellets. If you've tried any of these before just ignore it, but have a read through all the same.
1. Mixing pellets in with his normal food. This has to be done very slowly for him to accept it. Give him his normal food with a small spoonful of pellets mixed in. It's even better if he gets a mash of vegetables, because that way some of the mash sticks to the pellets. If he eats some of the pellets this way, great! You can move onto increasing the amount of pellets in his food over the next few weeks. If he won't touch them, try crushing them up and adding them to his food.
2. Soaking pellets in something. Carrot juice, apple juice, nectar or even a tiny amount of coconut oil might just get him to have a taste. If he's willing to try them, you can eventually reduce the amount of juice.
3. Smearing a little bit of salt-reduced peanut butter on one of his favourite toys or chewing materials, then sticking the pellets onto the peanut butter sometimes works to get a parrot used to the taste of pellets, and can make the switch easier. Shoving some pellets and peanut butter inside gumnuts can work wonders.
4. Feed high off the ground. Sounds weird, but parrots are more likely to eat if they are somewhere high. They feel safer, and are more likely to focus on the food rather than other things going on around them.
5. Pretend to eat the pellets. Parrots always want to eat what you're eating, and if Jet thinks he's missing out on something yum, he'll definitely try it.
Now throughout this whole 'weaning' period, you should have a small amount of pellets available at all times, just in a little food dish or bucket up high somewhere.
Also, have you ever tried training him? Something simple, like 'step up' or 'wave'? If you can get him to stop calling for even a little bit, you can reward his silence. If he 'waves' for attention instead of calling, even better.
Keep in mind it can be hard to train a behaviour like this out of a cockatoo, especially if it's been going on for a long time. This won't happen overnight, and is going to need the cooperation of everyone in the household. But, with luck on your side, I think you'll be able to do it.