BayJay2, private school converters are given a category of their own on page 8 of the Policy Exchange report - 21% in wave 1 (including e.g. Batley Grammar School and the Maharishi school), down to 4% wave 2, 5% wave 3, 4% wave 4 and not listed at all in waves 5-7.
Also, I guess there is a semantic difference between 'multi-academy sponsor' and 'multi-academy trust'. Similarly, a 'single academy trust' might be set up by a charity, teacher's group or parent/community group, all categorised differently by the Policy Exchange, so for the purposes of the report I think the Green School for Girls is a single academy sponsor that will be creating a multi-academy trust (like Waldegrave at the moment).
The way I understand it, roughly a third of new free schools are being created by existing or new education providers, a third by single sponsors (which I still believe includes academy converters and consortia), and the remaining third by third by charities/teachers/parents/community or faith groups. Any of these types might create a school with a religious ethos or denomination, although the Catholic church has ruled out being a sponsor to free schools.
It stands to reason that the most successful schools are run by sponsors/schools that have an established management team and track record, and the high profile flops are the ones run by amateur groups and private schools, or just bad chains like E-ACT.
There is certainly a growth in the number of successful converter academies sponsoring new free schools - that would also be a pragmatic arrangement for many LAs. But it would be perverse not to acknowledge that they were previously successful schools when run by LAs. In Richmond the limited number of new or relaunched schools maintained by the LA in the last 10-15 years have all been judged 'Outstanding' or 'Good' (I'm going to assume St Richard Reynolds is included there with Christ's, Marshgate and Kew Riverside). There's no evidence to support the exclusion of LAs from setting up new schools.