Aw, I'm sorry to hear about this!
Taming finches can be tough. Some are hard nuts to crack.
And unfortunately, you're right -- a tame bird can become less so when they're surrounded by other birds who are skittish. (It makes sense. If you walked into a room and saw several people looking very obviously fearful and terrified, you'd get nervous yourself. And if you saw that they were fearful of a specific person or situation, you'd probably adopt that fear yourself to some degree.)
That said, maybe you're going about it the wrong way?
Taming is tricky. You need to find out where they're comfortable and then slowly push them past their comfort zone, but you must do this without scaring them.
Let's say they're okay with you sitting alongside the cage or inside the aviary. Once they're really accustomed to this, you move to the next step, which would be placing your hand inside the cage with a piece of millet --- something where they can keep a distance but still associate that treat with you.
Gradually, you shorten the length of the millet and ultimately, they start eating out of your hand. And you just progress from there.
The problem, though, is that you can't frighten them. If they're comfortable with you sitting alongside their cage, and then you move forward and try to touch them, that's too much too soon. It's scary. So it becomes counter productive. We don't want them to associate you with fear. So it's about pushing them past their comfort zone in a gradual manner, without scaring them. If you see they're scared, back up a bit. Wait until they're comfortable again, then move forward.
I think that's the most common mistake -- trying to do too much, too soon. And you need to be consistent with the approach you take, so they know what to expect.
For instance, I start every session sitting alongside the cage for a few minutes. Then I open the door and put a hand in to offer millet. Then I put a treat in my hand and let them eat out of my hand. Then I'll handle them inside the cage. Then we progress to eating a snack outside the cage. Then to handling outside the cage. So they're eased into the situation. Every day we go through those steps faster and faster. But you can't just stick a hand in and handle them until they're totally comfortable (which can take a long time to achieve. And some birds never get to that point.) Sticking a hand in and handling a finch is unpredictable and unpredictable = scary. So you want to go through those steps, easing them into it.
So every session together, we go through the same steps. They like to know what to expect. The unknown is scary. We want to teach them that you won't do anything frightening or unpredictable. <--- that's really key. That should be your mantra. I want to show them that I won't do anything frightening or unpredictable.
My process:
http://www.petfinchfacts.com/all-posts/ ... her-birds/
Some general taming tips:
http://www.petfinchfacts.com/all-posts/ ... ing-birds/
Here's a writeup about using millet (or another treat) as a taming tool, just like Elana described:
http://www.petfinchfacts.com/all-posts/ ... cockatiel/
In most of my writeups, I chronicle my experiences with my cockatiels and budgies. That's just because I haven't tamed a finch since I started the site! But I'll chronicle that once I get a new "project". The process is really the same, regardless of species.
I'll caution you that taming aviary birds is more challenging. This is true of any animal. It's easier to work in a small space like a cage when taming an animal. The reason: a large area like an aviary gives them the ability to back away so they don't need to confront their fear/discomfort. In a cage, they don't have a choice. They must confront the discomfort and work through it. Confronting the discomfort is essential if they're to become tame.
Here's an example: we adopted a family of cats. The feral mama cat, her 3 kittens and her adult son.
The kittens and son did great because they were still young and impressionable. The feral mama cat was very frightened and resistant to taming. She lived in our house for about 2 years and we'd never touched her! She was very happy amongst the animals, but she avoided us humans. She would run if we got within a few feet of her. She had full run of the house so she could avoid us quite easily. So she never confronted her fear or discomfort; she just avoided us. We'd hope she'd lose the fear once she saw that every other cat in the house was tame and enjoyed us. It didn't work like that.
Then the mama cat got sick. We had to isolate her in the bathroom so we could monitor her and ensure she got her medications in her food and so forth.
Something incredible happened while she was in the bathroom. She was forced to confront her fear of us. She couldn't just run away anymore. So we kept her in the bathroom for a couple weeks after she recovered, working with her intensively and spending a lot of time with her. That's what it took to tame her. She had to be forced to confront that discomfort and work through it.
The same is true of many other animals, including birds. They generally avoid the fear and discomfort. So you might find that working with them in a cage or a smaller area is easier.
I would also focus on 1 or 2 at a time. Working with 6 at once, plus 2 canaries is difficult!
