I don't think it would harm her to start at 6, but equally I don't think it is necessary or that any later starters are doomed musically speaking.
Unfortunately, many of the children I know who have started very early (7 or under say), have given it up within a few years or even sooner, whereas those who have started instruments aged 10 or over seem more enthusiastic and committed, and to progress very quickly indeed.
Two of mine play instruments so far. One is almost at Grade 8 standard after just over 2 years, the other about Grade 5 standard after a year and a half.
Someone said shared lessons are a waste of time. Well, yes and no. One to one is definitely better, but often group lessons let the child get a feel for an instrument at low cost. For example, at our primary school lessons are £40 a term, usually shared although my son was the only one in the school playing that instrument so he still got an individual lesson. At secondary school, there is a huge leap to £120 a term, albeit one to one.
Everyone here seems to recommend practising before school - they obviously all live in detached houses or have deaf neighbours! My children can play the digital piano using headphones, but I really don't think a burst on the saxophone would go down well at 7.30 in the morning!