First time owner with a skiddish and possibly traumatized budgie, I'd really appreciate some help🙏🏻

I've been posting here a lot sorry🥲

This is my bird Periwinkle,or Peri for short. My mom got her late November and then kind of abandoned her 'because she wasn't as fun as her childhood birds', so 2 weeks ago I moved her to my room to hopefully improve her life, and I have some questions about training, the current setup and a possible new cage. If anyone could answer even just one of these I'd highly appreciate it🙏🏻🙏🏻

Training: The issue with training is that Peri is super skiddish and I suspect this is because my mom accidentally traumatized her on day 1. Peri escaped so my mom had to chase her, which she obviously didn't like. She also spent her first month in the living room where my brother plays video games, so she had to endure A LOT of screaming. So yeah, it's no surprise that she's taking a long time getting used to me. I've been following tips I found on YouTube for training skiddish/traumatized birds, but none of it seems to work. I've been putting my hand in the cage and offering her millet from both inside and outside the cage for several hours a day since moving her into my room, but no luck. She freaks out whenever I try to interact with her or even get near the cage.

Four days ago however I had some kind of breakthrough I think? While grabbing her food bowl she shot past my hand and flew around my room. I didn't want to traumatize her again so I took the big perch out of her cage and tried to get her to land on it. Turns out she doesn't mind me holding the perch she's sitting on, as long as she's on the opposite end of where my hand is. I'm proud of her honestly, and it also made me wonder if this is the key to earning her trust. For the past four days I've left the cage open because as you can see in pic 2 it's quite small. She got used to the perch really quickly and I can walk around my room with her on it, but still can't get closer to her. How do I proceed from here? She does eat millet so I tried to reward her with it, but she refuses to eat it out of my hand unless I corner her and shove it in her face💀

Current setup: Please ignore what's in her cage, she has more toys in and outside her cage right now. This is more about the placement and hoodie. There's a small air vent right above my window that I can't close. During the day Peri just goes wherever she wants, but to protect her from the air flow at night I put my blanket hoodie on the cage like in the pic. Is that safe? Do I need a better cover? Is this even necessary at all?

New cage: My current cage is too small so I'm looking for a bigger one. I really like the one in pic 3/4. Just wanted to know if it's appropriate. I'm planning to leave it open 24/7 eventually. I'm also planning to get Peri a buddy once she's trained and stuff, so the cage needs to be big enough for two birds (I'll worry about things like a second cage later)

Alright thanks for coming to my TED talk😭 sorry for the amount of text-