I assume you don't contend that academy chains are not run on a business model
Any large charity is run as a business. However, a charity does not have shareholders. Any surplus must be retained within the charity and used for the purposes of the charity. Unlike a business a charity is not generating profits for its owners.
the fact that an academy chain is a charity is no bar to connected businesses making money from them
It does not completely prevent a connected business making money from it but it severely restricts the ability of the business to do so. It cannot simply syphon off any surplus. It can act as a supplier to the charity but the trustees must, if challenged, be able to show that any payments are in the charity's best interests and reasonable for the services provided.
nor indeed is it any bar to commercial salaries being paid to chief executives and the like
A charity can indeed pay its employees whatever it wants. However, there are restrictions on its ability to pay trustees (the equivalent of a company's board of directors) and people or businesses associated with trustees. Unlike company directors, trustees cannot in general be paid just for being trustees. They can be paid for doing other work for the charity under certain conditions but only a minority of the charity's trustees can receive such payments.
No academy chain operates out of philanthropy
I would dispute that given that one academy chain was set up by someone who has a long history of philanthropy in the education sector, most of them are independent (i.e. not related to any business) and none of them distribute their profits.
Have you ever worked for a large charity? Lots of corruption and high salaries
I have never worked for a large charity but I have been closely associated with a number of charities of all sizes. Yes, the chief executives of many large charities receive high salaries but generally below the salaries paid to private sector chief executives controlling organisations of similar size. As for the "lots of corruption" allegation, all I can say is not in my experience. I won't say all charities are squeaky clean but there is no reason to believe there is any more corruption in charities than there is in the public sector or the private sector. I have encountered corrupt individuals in charities. That is not the same as saying the charity itself is corrupt. There may be corrupt charities but I have yet to encounter one. There was a thread on this subject recently.