It seems like they do just keep trying. Here you buy finches at the pet shop and they are generally young adults by then. And it depends where they come from as to whether they are used to people or not. My Java finch adults in the aviary are very highly strung and wild. My society finches aren't as wild but still cautious of us. Our Bourkes parakeets that have been raised in our aviary are 3 of the cheekiest little things and are constantly landing on me to check me out, flying around me, showing off. I think it helps they are 3 little boys. They are tamer than the two I hand reared, though I think those two have bonded with each other being a boy and girl.
We had a ring neck dove fly in one day and after quarantine she joined the aviary birds. They were all terrified of her to start with as she's so much bigger than they are, but things seem to have settled down now. But, though she doesn't like hands, she takes great delight in landing on heads! The young Bourkes have copied her and are now very confident about interacting with me.
I know some people hardly ever go into their aviary, but for me, most of the pleasure of having birds is interacting with them.