Only if your definition of Star Trek excludes some Star Trek (on, as you note, the basis of speculation over what Gene "really" wanted to do).
It's not about "excluding," it's about recognizing the overall thrust of the whole. Something that's mostly striving for believability and occasionally has sillier side trips is not the same as a series that's constantly, deliberately turning the absurdity up to eleven. If
every episode of ST had been in the same vein as "The Trouble With Tribbles" and "I, Mudd," then it wouldn't have been the same show that it was. It's a question of proportion.
And it's not speculation that Roddenberry was trying to make a departure from the campy SF of the '60s, to create something more believable and adult. It's documented in many places, including the TOS series bible. The very first thing the bible contains is a 3-page treatise on believability, telling the aspiring writer for the show that even though it's set in the future, the writer must still apply the same standards of believable characterization that would apply in any contemporary drama. (Amusingly, it cites something that actually happened in "Balance of Terror" -- Kirk hugging Yeoman Rand on the bridge during a crisis -- as an example of an unrealistic scene that should be avoided. So they were aware of their mistakes and tried to remedy them.)
AND SO, IN EVERY SCENE OF OUR STAR TREK STORY . . . . . .
. . . TRANSLATE it into a real life situation. Or, sometimes as useful, try it in your mind as a scene in Gunsmoke, Naked City, or some similar show. Would you believe the people and the scene if it happened there?
IF YOU'RE ONE OF THOSE WHO ANSWERS: "THE CHARACTER ACTS THAT WAY BECAUSE IT'S SCIENCE FICTION", DON'T CALL US, WE'LL CALL YOU.
True, the bible stressed believable characterizations over believable situations; the producers of TOS were okay with outlandish situations so long as the characters responded to them in a believable, naturalistic way. Still, the intent was explicitly to make a show in the same vein as the respected adult dramas of the day.
Gunsmoke was heralded as a realistic, adult Western.
Naked City was a police drama renowned for its intelligent writing and semi-documentary filming style. These are the kinds of shows that
Star Trek was intended to be like. They both had humorous scenes and episodes, but it would be totally wrong to say that an outright comedy like
Blazing Saddles was "
Gunsmoke-like" or that
The Naked Gun was "
Naked City-like."