The LEA is one part of children's services, which is one part of the council. It uses specific funding to provide services for state-educated children. The funding provided is specifically for state-educated children, and there are really clear accounting practices to ensure the money is spent on what it is intended for.
There are other parts of children's services which provide other services, such as safeguarding. Some will be part of the council and part of children's services but not part of the LEA. Some funding is for services which are available for everyone; others are restricted to certain populations.
Accounting is a way of making decisions, through budgeting, and the making sure those decisions are adhered to. It's not a trick - it's deliberate. It's just that most decisions about limited resource allocation mean some people get things and some people don't, or get it later.
Education isn't fair. It's inherently unfair that some children are unable to access mainstream education due to SEN. Some local authorites have better alternative provision. Some parents are able to access private provision because they can afford it. Some parents are able to access better provision because they can navigate systems. It's really unfair that some people have a lifelong disadvantage because they can't access the right education as children. It just is unfair and it's wrong.
If parents are able and choose to opt in to private education, they are choosing to opt out of the LEA provision. In this case, that includes LEA-funded OT (but not health-funded OT). I wish they had better choices available.
The alternative is to campaign for a return to centralised services, where all spending is dictated by Whitehall, and everyone gets an identical offer of services, and there is no private education or healthcare - that would be fair.