I think you are confusing the CPTP with the BFTS (British Flying Training Schools) scheme which started in 1941 and which used 6 Civilian flyng schools, but with the syllabus overseen by the RAF.
The 6 schools were:
1 BFTS Terrell, Texas
2 BFTS Lancaster, California
3 BFTS Miami. Oklahoma
4 BFTS Mesa, Arizona
5 BFTS Clewiston, Florida
6 BFTS Ponca City, Oklahoma
My late Father trained at the Spartan School Of Aeronautics in Miami, OK, Dec 1943 to Jun 1944.
The Arnold scheme was hosted by the USAAF. There was a similar scheme, the Towers scheme, hosted by the USN. My late Uncle trained under the Towers scheme at Detroit and Pensecola.