I kind of wanted to reach through the screen and spot her on bars, OzZee, and the headstand and BHS made me nervous. She's cute as a button and very strong, but the video was scary to me. But I hasten to add that I don't know a lot about neuro-skeleto-muscular child development. Interesting that her big sister appears to be one of those 8 year old L8s. My DD was put on pre-team as a seven year old, shortly after taking her first rec class!
USAG doesn't let them compete in sanctioned meets until age 6, but other systems (like AAU) let them start younger.
I don't know for sure, but I'm pretty sure no one in my kids' gym is doing any back handsprings before age 5, and more usually closer to 6. The children I mentioned in my earlier post all progressed very quickly once they were on pre-team; one of the boys actually went straight from rec to team. The three-year olds in the gym are definitely learning some gymnastics skills, but it's more along the lines of forward/backward rolls, walking/hopping on a low beam back and forth, learning pullovers with a solid spot, etc. The preschoolers do learn some trampoline progressions with six or seven steps (remembering back to when DS was still in rec).
How common is stuff like the posted video? I don't get the sense that it's super common in my area.