I also thought he never reported meeting Cochrane and whatever happened to Nancy Hedford.
Indeed - the title of this thread should perhaps be "Do you think Starfleet would have felt Kirk had failed any of the missions if Kirk actually told them the truth about those?"...
Kirk lies to his superiors in the very first episode already, refusing to tell Starfleet the truth about the demise of Gary Mitchell and Elizabeth Dehner. Apparently, it isn't too difficult to erase or edit logs after the fact, either - and if the logs don't involve incidents of note, nobody will go through those "Court Martial" style, supposedly-difficult-to-forge flight recorder tapes and find out what really
Actually, forging those tapes may also be simpler than we think. It was said in "Court Martial" that editing could be done only by Kirk, Spock, and Finney. Now, the latter two are specialists - but Kirk is a generalist, so forging of tapes should not be a "skill" issue since he presumably would lack the skill. It's more probably an "access" or "authority" issue, and if Kirk does have the access and the authority, then no adventure is safe from his creative reinterpretation.
As pointed out above, Kirk has obfuscated the deaths of several people quite explicitly. The disappearance of Marla McGivers could well be taken as an implicit fraud, then.
"Hey fellas, there is this 300-year-old genetically engineered tyrant on that planet. All of you secret admirers of his regime, go find him and take him home."
This would imply that Khan admirers are a disliked minority. But Starfleet apparently consists almost exclusively of neo-Khanists, unless Kirk's own ship for some strange reason was an exception to the rule. If Kirk reported about Khan's survival, Starfleet would probably send the Federation Flagship and beam down an infrared carpet so that Khan could start his triumphant return with suitable pomp.