I've been involved in a number of organizations ("clubs" mostly) which somehow have to combine volunteers with people who make a living out of it. I find that most people struggle to keep the club atmosphere, where a lot of things are done by volunteers, for the members, combined with paid services.
My flying club is a club. It is run by a volunteer board of directors. All the aircraft are directly or indirectly owned by the club. The club (a large one: something like 600 members) has a few paid staff who manage the reception and do a lot of other admin stuff. This obviously includes invoicing the hiring of the aircraft, landing fees at home base etc. The rental rates for the aircraft are public, as are the total number of hours flown and the cost associated with running an aircraft (fuel, maintenance, insurance, ...) so people can verify that the club indeed offers the aircraft for the lowest decent rate possible.
All flying instructors are nothing more than members of the club, as is the CFI. The club, as a courtesy, provides a feature in the internet booking system that when you book a club aircraft, you can also book an instructor, if the instructor has identified himself as 'available' for that period of time.
Payment for flight training is done to the instructor direct, although every member is free to keep an account at the club and the club, again as a courtesy, allows you to deposit money into an instructors account if that's what you agreed with him/her. But for all practical purposes the instructors are freelance, with no labour contract between the instructor and the club.
The CFI, instructors and the board of directors have gone through the hoops of getting the club registered as a "registered facility" for PPL training. A few instructors including the CFI have also created a separate organisation, called the FTO, which is certified for CPL/ME/IR training, but I don't know the specifics about this.
The maintenance organization we have is a separate "BV" (Ltd. in the UK), which is wholly owned by the club, but financially separate. They have paid staff and the club pays the maintenance organization for any work done. They also maintain a few non-club planes, and for historic reasons some club aircraft are maintained by another, commercial outfit on the field.
The bar is contracted out to a caterer.
A few members have aircraft of their own, which they make available for club use (typically for the complex and multi part of CPL training). As a courtesy, these planes are included in the reservations system as well, but the rules on usage/checkout etc. are set by the owner. The club, as a courtesy, will handle the invoicing for the plane rental, and I assume the club charges a small fee for that (on the order of 5 euros per flying hour or so).
All in all I think my club has struck a good balance. I have heard of clubs gone under because they were run as a commercial school by the CFI, leading to the situation where fresh PPLs would run away because they were not, for all practical purposes, welcome to rent planes there anymore - because the CFI did not make money from rental. I have heard commercial schools go under because they were too expensive. And I have heard clubs go under because they were not run professionally enough, leading to the situation where instructors could not count on having a somewhat decent stream of business and as a result offering their services elsewhere. Or the situation where flights were wrongly invoiced or not invoiced at all because the reception staff was made up of club "volunteers", who were forced to do reception duty once or twice a year.