It doesn’t matter if what the kid said was way worse than what the mum has admitted, or if she has awful parents. Even if the body cam video shows her saying horribly abusive things, this is not an appropriate way to deal with it. She was not a physical threat and couldn’t be questioned until she’d sobered up anyway. They should have left and asked her to attend the station the next day with her mother. The arrest was punitive, it served no purpose in terms of investigation or public safety.
I would have thought that the Child Q case review would have led to police forces really looking at how they deal with vulnerable children, but sadly it appears not. Her mother should have been her appropriate adult.
It sounds like, like many 16 year olds, she lied to her parents about going to a sleepover and then snuck out to join her older sister. That happens to lots of parents of 16 year olds, whether they know it or not. But even if her parents were neglectful and letting her do it, that wouldn’t make treating her that way OK. Kids with shitty parents need more protection, not less.