Agreed!
Intelligence definitely varies among species and there are different types of intelligence too.
Finches aren't like parrots and more interactive hookbill species -- they can't learn to speak and they don't "play" like many of the smarter species. But that means they also don't have many of the behavioral/emotional problems of those larger, smarter birds.
95% of finch behavior is driven by instinct, I think. Whereas some of the smarter bird species are less instinct-driven and more "conscious."
But like all animals, they do best with routine and can easily learn through repetition. So for instance, my finches (also Zebra finches) know that when I come in and turn on their light in the morning and start pulling out the food bowls that it means it's time to eat and come out to fly! So they get excited and start bouncing around and singing.
I've found that finches do better learning when you associate a unique whistling or similar sound (rather than words) with a routine. I'll make random whistlings and squeaks until one resonates with them (they all fall silent and look at you or all start vocalizing in response).
Then, I'll pair that sound with an action, like feeding time or putting my hand inside the cage to clean or whatever.
I guess it's my attempt to speak bird. Over time, they'll recognize what your sounds mean. So for instance, if I make the feeding time sound, they get excited and start making their way to the lower perch.
This works better with the smarter hookbills, but it also works with finches as long as you're consistent.
From what you wrote, it sounds like you've tried to let her out of the cage?
This can be tricky, as catching them can make this more trouble than it's worth. But if you have a room that you can bird-proof and don't mind catching her or waiting for her to return to her cage, she'll love stretching her wings.
Just be sure to eliminate dangers in the room. NO CEILING FANS! These are a death trap! No toxic plants. Cover trash cans. Avoid those little air fresheners with the sticks in the jar -- they're full of chemicals and they're the perfect size to land on. And just be sure to create an environment where she can't escape to the outdoors.
I have a bird-proofed room for my guys and I'll pull up the food bowls for disinfecting and cleaning while they're flying around. Then, an hour or two later, I'll return them when it's time to go back to their cage and they make their way back to the cage naturally because they're hungry and I've stocked it with goodies!
Many finches aren't hand-friendly if they haven't been handled as youngsters. But with training and patience, many adults can be hand-friendly. The cliff's notes version: You can start by moving the cage alongside you while you're on the computer or doing something else where you're sitting quietly, with minimal movement. They'll quickly get used to your presence. Then, you can stick a finger through the bars or put your hand through one of the doors and just be patient. They'll come to check it out. Stillness is the key early on, then gradually, you start moving slightly and get them used to that. Once they're comfortable with landing on your hand inside the cage, you can try it outside. As with any animal, it's a gradual process. And it won't work with all birds. Some just aren't all that interested in humans.
Did you get her a friend? I *strongly* recommend this. Finches are super social. In most cases, a single finch is a sad finch.
I think of it in human terms: I love my animals, but no matter how bonded you are and how human-like an animal is, it just doesn't compare to the company of another human.
I think it's the same way with birds, dogs, cats, etc. A relationship with a human is wonderful, but it can't replace the relationship with another of their own species.
If you're still covering the cage, make sure it's a fabric with a tight weave that her nails won't get caught in.
Years ago, I covered my cage with a fleece blanket and one of the finches landed on the bar and her nail snagged the fleece fibers. She got stuck and I had to rescue her. She wasn't injured, as I was in the room when it happened and helped right away. But if I wasn't there to help and she remained stuck, she could have been hurt or even died.
I don't cover my cages anymore. They have a quiet room that's nice and dark at night -- no TVs or lights to bother them and no foot traffic to cause night frights.
Best of luck! I saw you posted a few weeks ago so I'd love to hear how your girl is doing!