Any local council thinking of setting up a charity should ask two important questions. Firstly, is the purpose of the proposed charity to perform a function that falls outside the permitted powers of the council? (Grants to individuals would be the obvious example). Secondly, is the purpose so valued by the community that it will attract thousands of pounds of individual donations each year? If the answer to these questions is no, don't set up the charity. The project could be run from within the council. We already have one charity for every 350 people in England and Wales (or one for every 150 households). Do we need any more?
Funding relationships between local councils and their charities are still subject to public scrutiny through the local government transparency processes and, unless GPoC is held, subject to the S137 limit. Councillors who serve as Trustees of these charities tread a fine line as far as their conduct is concerned.
I haven't mentioned CICs. It's not my area of expertise, but, in essence, a CIC must be a limited company and can't be a charity, although a charity may set up a CIC as a trading subsidiary. A CIC can pay directors and shareholders, carry out permanent trading activities, retain profits (but pay Corporation Tax on them), can have just one director, not entitled to rate relief, report to Companies House, not Charity Commission.
Finally, reconstituting an existing charity with a new legal structure doesn't involve a public process if the objects are to remain the same.