Rightly or wrongly, the RAF has decided that merely because a flying school is CAA approved (or a BGA club similarly) they still have to audit it. This means that there is no blanket permission to use either.
To be fair to them, the RN does the same with RYA centres - I am issued with an 'approved list' I can use for cadets.
Now, whether a system of approval of civilian FTOs would be cheaper than the current system is worthy of debate.