The story that the Wright brothers did not want to show their plane because they did not have a patent is a pure myth because:
1) The May 22, 1906 patent was for a glider not a powered airplane and offered no protection regarding the propellers, engine, transmission chains, etc.
2) Everything relevant in the brevet had been already published by Octave Chanute in Aug. 1903 in L'Aerophile and replicas of this glider with of without modifications had been tested thoroughly in France during 1904.
3) The only thing that might have been protected by the patent (at least according to what the Wrights believed) had been the wing warping (ailerons, roll control) which would not have been visible on pictures with the plane flying.
4) Anyway, the patent was published on May 22, 1906 and there would have been no logical reason after that date not to show, at least, photos with the plane flying, to gain credibility.
The Wright brothers were bluffing. They did not have any workable airplane before 1908.