I use a king sized black bed sheet. Cost me about 10 bucks at walmart. I feel very comfortable using the sheet cause I'd sleep under one myself. The fabric breaths so I know I won't be suffocating the bird under it. When it comes to black out fabric, I don't know. Maybe it's safe but perhaps it's too heavy. The goal of covering the cage at night is to give the bird enough darkness to sleep and block drafts, but it doesn't mean they need 100% pitch darkness.
In fact, I have a plug in LED nightlight near the bird cages. Making it too dark can actually lead to accidents like falling off (especially for a young parrot). My sheets aren't quite big enough to cover 100% of the cage. Only about 90%. But that's ok cause I face the slightly open end toward the back wall where the night light is. This way they could still see in "moonlight" in case they really needed something at night. With this set up, the parrots awake in the morning the same perch I left them at night so I think it's plenty dark enough. But I do keep the window shades closed to prevent sunlight beaming them in the morning.
As for uncovering. I wouldn't do it too quickly the first week, but beyond that, no problem. I just rip the sheet off the cage in one motion as quickly as I can cause the parrots get really excited to see me. What I mean by slowly the first week is making it a slow steady motion so that it doesn't frighten the parrot by moving too quickly.