I think you know that the rules for feeding captive iguanas in northern latitudes (North America and Europe) exclude all animal protein. That is because for these particular iguanas, digesting a mouse is hard on their kidneys.
The problem we have is the near impossibility of humidifying an iguana's living quarters to 80%RH. The highest I've ever achieved was 60%, and most captive iguanas live at 30 - 45%RH. That causes the iguana to dehydrate with every breath. Dehydration causes kidney failure over a period of years. Most of our iguanas die young, seldom making it very far into their teen years. Most die well before they are ten years old.
It is obvious to us that Green Iguanas have a strong appetite for crickets, mealworms, and mice. I agree with you in that I think one would instantly attack and swallow a parakeet if it could.
We as a group, through much unhappy practical experience, through blood tests in vet care have connected early death with dehydration which kills their kidneys. So we react to that by avoiding any amount of animal protein in their diet. This gives their kidneys a better chance. Captive iguanas get all the nutrients they need with leafy greens and starchy vegetables, like cooked peas and squash.