The Enterprise has never been referred to as the or a flagship in TOS or the films. This flagship business started in TNG. In TOS the closest we came to seeing a flagship was in "The Ultimate Computer" wherein Commodore Wesley aboard the Lexington is in command of the wargames squadron.
It wasn't the Flagship, but it was repeatedly stated that the Enterprise was a special vessel and there were only 12 like it in all of Starfleet. At least until the movies where it's status as a aging vessel was a plot point. The Enterprise-D being the Flagship was more or less just an extension of the original's status.