I have not heard any issues with independent flight instruction at my home 'drome
Check to see if your local airport has a document regarding "minimum standards for commercial operations". Typically these are written with specific language to prevent freelancing CFI work by imposing onerous requirements (facilities must include pilot's lounge, flight planning area, student training area, public telephone, public restrooms), hours of operation (e.g., 8-5, 7 days a week), minimum number of aircraft available, etc. At my airport, they require several hundred square feet of building (leased, of course, from the airport), two airplanes for rental one 4 place and IFR, one full time CFI, open during normal business hours, pilot lounge, restrooms, instruction area, flight planning area, parking area, and public phone. It does not allow a 'through the fence' operation, e.g., you cannot have a building next to the airport and do all your instruction and pilot training there and then walk over to the airport. everything must be on site and rented from the airport.
These documents are often online when the airport is owned/operated by a port district, city, county, etc. I have read some that specifically exclude flights that originate and terminate at other airports; which allows for landings at this airport. however, others do not have that exclusion so, technically, go do a dual-instruction touch and go at an airport and they can cause you all the grief they want.
Want to perform flight training at the student's location? An existing FBO just needs to call the airport manager to have any freelance CFI on the field kicked off, shut down, lose hanger privileges, and served with legal papers. The framework of the document is handed down from the FAA so do not expect to take up any 'restricting free enterprise' complaints with them. You will likely find associations like the AOPA are firmly on the side of the airport management and larger FBOs.
Yes, these are public airports taking public money.
Yes this is total impeding of small business interest.
Yes, this the very definition of restricting access of public property to prevent competition.
yes, the airport is picking who gets to play and if you are not in with the airport authority they will boot you off the airport.
yes, this is blessed by the FAA
No, they do not care if anyone complains about the fact that this is private-government collusion to inhibit competition.
here is a link to a recent survey of the typical requirements that airports include in their minimum requirements document.
https://www.nap.edu/download/14491