There you are, in a nutshell.
Now an ATC Bring and Buy Sale, Car Boot etc at your local aero club and blag a flypast or too.
Flying Display pilots require approval or a licence to display at an event and have to follow military or civil regulations. OTOH any licenced pilot can make an approach and departure at your airfield.
An off-field event would require a special licence from the CAA for civilian aircraft as, by definition, they would otherwise be doing unauthorised low flying,