Sounds less age-related and more cage-related to me. I apologize if this sounds harsh, but imagine a human stuck in what amounts to solitary confinement for twenty-odd years. That would be enough to make anybody a nervous wreck. A bird's cage should be like a child's bedroom - a place that's safe and comfortable and all theirs, where they can sleep or play or hide, but it should never be a prison.
You have a couple major positive points, though:
1) "Flying around the room" sounds like she has all her feathers and the ability to fly, which is very, very good. Flying lets her avoid scary things, which makes those things much less frightening. Imagine you're out on the ocean and you see a giant shark. You'll feel much safer in a boat than in the water, right? Birds feel safe up high and far away. From there, they can observe everything and maybe, if they decide it's not so scary after all, they'll decide to eventually come down for a closer look. Letting them run away paradoxically encourages them to come closer in the long run.
2) There's another benefit to all that flying. It's a workout. It will help her get out nervous energy, improve her sleep, appetite, weight, muscle tone... basically all the benefits you'd expect from starting an exercise program, plus it's extremely mentally stimulating.
3) You're making an effort to change her life for the better and have come here for advice.
My first thought would be to open the cage door and just leave it open. Let her come and go as she pleases. (And if she refuses to come out, that's okay - what's important is that she has the choice). Make food and water available inside the cage and on the playstand, and make the playstand food the tastier option. Then, ignore her. Go about your life as if she's not there. If you see her watching you, acknowledge her with a gentle greeting, "oh, hello there," and then continue with whatever you were doing. If she shows signs of fear or aggression (which, in a timid bird, is basically just another way of showing fear), either remain still (if you're already still), or move away and give her some space (if you're in motion). Trying to control her is futile. She needs to learn that she can control YOU. Controlling you means she can interacting with you (or not interact) on her terms.
That's the biggest hurdle. Once she realizes her behavior can influence your behavior, all you have to do is show her that you have some WONDERFUL behaviors she can create. You are a treat dispenser. You're a giver of scritches. You're a fun person to talk to. And all she has to do to get these things is ask! Asking, of course, can take many forms. It may be simply sitting nearby or making a particular noise. It may be stepping up or otherwise doing some simple task.
Whatever steps you take, just be aware that nothing will change overnight. She's had 20 years to develop the behaviors she has. It may take weeks or months or years to see significant changes.
To give you an idea what I mean: I have an 11 year old TAG who came to me fully feathered, yet unable to fly - I suspect she was clipped early and simply never learned. She came to me in mid-April. Tonight, for the first time, she did a flapping jump to get over an eight-inch gap. It's the first time I've seen her trust her wings even the teeniest bit.