I’m sorry that you and your family had to experience the loss of your cockatiel that way, and wish the vet had told you to take your little friend home instead.

Regular vets who aren’t avian specialists really can’t do much for pet birds aside from mending injuries; an avian specialist can help you diagnose diet issues before the bird is critically ill, so it might still be worth taking in a bird that seems a bit under the weather to see if it is a diet issue - an awful lot of pet bird problems are diet issues. I took mine to one to have a microchip implanted before exporting her to Germany; not a huge deal with big parrots (they can be injected into the wing), but for small conures, it involves general anesthesia, so I went out of my way to find someone who had a lot of avian experience.

But yes, once a pet bird appears seriously ill, it’s likely too late to do anything other than keep it warm and in a place it feels safe, and let nature take its course.

As far as I know, pretty much all animals, especially prey animals, hide illnesses and injuries, and that is definitely the case for parrots.