I am just giving this one more go, then I will give up. Local authorities have no control over the salaries paid by "local authority schools". Schools set their own budgets, they control their own HR, etc., etc. That is ALL schools, not just academies. "Local Authority schools" may buy into - or not buy into - the back office systems of the local authority, but whether they do or not, the school determines their own policies. The local authority does control its own employment - the core functions - which, in relation to education are limited to:
1 Back-room services (eg administration of payroll, legal requirements, brokering contracts for, say, IT services)
2 Co-ordinating admissions to all state schools in the area including free schools and academies.
3 Responsibility for children with special educational needs (SEN) and education welfare services.
4 The legal duty to manage the supply of school places.
The entire of the rest of the monies - and the spending of those monies, policies etc - lies with the school. Even if the payroll is administered by the local authority, and even if the staff are technically employed by the local authority, that is a paper exercise and all control lies with the school, with the sole exception that the local authority takes part in the selection of the head teacher. The local authority has no other powers or responsibilities.
If you want to blame somebody for the situation, the people to blame are the government who created this situation. Local authorities have nothing whatsoever to do with it.
That is the reason why the way that local authorities employ their core staff is entirely and totally irrelevant when it comes to the way staff are employed in schools. You may feel hard done to about that. But continuing to blame the local authority for paying more than your school is unreasonable and ridiculous. And whilst I understand that HR / employment may not be your field I am rather surprised to be explaining to someone rather senior in admin and finance how their employment is funded and managed. Because despite claiming that you know how things are you are showing a remarkable lack of knowledge on how things are.