@noblegiraffe
I'm not entirely sure what jurisdiction councils have over academies. I thought we reported directly to the DfE.
As far as I'm aware LAs have no direct jurisdiction over academies in terms of education but their public health remit covers everything that happens in the borough. E.g. in 'normal' times the LA can shut down food outlets that are the source of food poisoning outbreaks and they can intervene at any premises that are an epicenter for infectious disease.
This is a public health issue. If an academy does not have effective measures in place to contain covid then that affects everybody in the borough, whether they have a direct connection to the school or not. We know that schools in general are driving community spread and LAs have a public health duty to everyone in the borough, including those studying or working in academies.
I'm not saying this will definitely be successful but it's another legitimate way of raising the issues and pushing for change in schools more generally. Sometimes a 'bottom-up' approach can be more successful, especially if all you're getting back from central government is a load of slogans.
Probably you'll be frequently fobbed off with, 'Academies? Nothing to do with us!' So you might need to frequently reiterate, 'Public health! Everything to do with you!'
Worth a try?