those must have been big cages to have more than 10 birds per cage! Were they aviaries? They would have to be.
I'm like you, I am hoping I will have better luck with one bird per cage in the taming process. I hate to do it but I might even move one to another room so he or she can focus only on me for a bit, then maybe switch. But not sure if that will help.
For now I can have my hand in the cage and Chirp will step up to my finger pretty good and Pips will step right up to a perch. Keeping them in cages for now 'till their flights are in which will be harder on me than them probably!

We have no avian vets here either.