I'd be interested in who provided the copy for the PR release, because I believe the journalists have got this story wrong.
I worked as a supply teacher for a while, including for Capita, who use a payroll company based in the Channel Islands.
The way I remember it working is:
Capita pays considerably 'below scale' ie the amount a qualified teacher should receive on a daily basis if paid directly by the school.
Capita then persuades workers that they won't lose out by being paid through a payroll company.
Technically, the supply teachers are self-employed, which means they can claim expenses, such as travel costs, lunch allowances, office and telephone costs against tax. In my case, this nearly always brought my tax liability to zero. (It should be noted that I wouldn't have paid tax anyway, because I didn't earn enough, so it was a con.)
Next, the supply teacher has to pay the employee's and the employer's national insurance contributions.
On top of that, the supply teacher has to pay the payroll company an administration fee.
The end result is that the supply teacher earns less than if he/she were being paid directly, but the payroll company and agency (in this case Capita) benefits. Needless to say, I only worked this way on a few occasions, before telling them to * off.
I'm glad that the government has finally taken action on this. It wouldn't surprise me if Capita has issued the press release and journalists haven't checked it out properly.
I don't know why there is a discrepancy between state and private schools. Maybe HMRC has decided to stop the abuse by targeting the agencies through the schools. They could control who state schools employ, but not private schools. However, that's a guess.