A child of 16 is entitled to live wherever they want. If they leave home they won't be forced to return.
A child of 16 is entitled to apply for a school of their own choice.
If she had applied to a state school they could not legally have excluded her on the basis that they weren't happy with her living alone.
Whilst the rules are different for independent schools, I think the independent school was absolutely right to admit her. The living arrangements of their sixth form pupils are none of the school's business.
The school has a general duty of safeguarding for all pupils. As far as I can see the only negative effect that has been suggested so far is that her grades suffered. That is not a safeguarding issue. If she was a victim of abuse and the school spotted signs of this they should have taken appropriate action. Similarly if others were being abused due to her living arrangements the school should have got involved. But I don't see any evidence that either the girl involved or anyone else was actually abused.
The school was under no obligation either to admit her or to create a situation where other DC were vulnerable because one of their classmates could host anyone anytime she like with no adult supervision. Schools have a duty of care
If the school had previously admitted boys who were living alone they would potentially have been guilty of sex discrimination if they had refused to admit this girl.
As for the second part of this sentence, words almost fail me. Sixth form children do things all the time without adult supervision. They also take advantage of parental absences from home to bring their friends round.
The fundamental is that there is absolutely nothing wrong with a girl of sixth form age living alone. I can see no reason why the school should have refused to admit her and no evidence in anything that has been posted that the school has done anything wrong or that it has failed in any way.