Thats a pretty good system that you describe there Ghenghis. I assume the aircraft is the equivalent of the old public transport regarding maintanance and engine life in order to be leased into the club?
That is correct.
The OP is of course going to have the problem of finding a club as enlightened as yours that will allow effectively a competitor to the club and its instructors under the blanket of the clubs approval. I would not let you do what you describe under our approvals as I could not be assured that you would meet our standarisation and quality requirements but as overseeing CFI I would still be responsible for you. You will find a lot more schools that will give you the same answer. As they will basically see you trying to ride on the back of their approvals without having done the legwork.
When I or the other instructor in the syndicate teach on our aeroplane, we're under supervision of the CFI, working to his standards, using his training records, and he flew with and approved us before we were allowed to teach on it; and the club charges something for all this - you get the picture.
And everybody agrees to work to the club FOB as if they were normal renters. The only two real differences for syndicate members are (a) much cheaper flying, and (b) no real restrictions on how much or little we fly it in any booking (apart from a maximum half day per weekend per member limit, unless agreed by the syndicate and club, which is to ensure some revenue at weekends for the flying school).
It works, and I doubt you'd be uncomfortable with it Bose.
Yes, it is a very enlightened club and I enjoy being part of it. It's small, but friendly and competent.
G