A 34 year old seduced a 19 year old in order to get to her father. And that's just one example.
As a hero of his time though? Awesome. Modern lens? Problematic.
It's just one example because there aren't very many others. And with respect, it's inapt. Kirk suspected Lenore's involvement from the beginning, but the script and Shatner's acting choices clearly show that Kirk actually had feelings for her. I realize that the prevailing interpretation of "Conscience of the King" is as you describe, but it's really not a fair reading of the dialogue and Shatner's performance. And the entire final scene undermines the idea that Kirk was toying with Lenore:
MCCOY: Medical report. (hands it over) She'll receive the best of care, Jim. She remembers nothing. She even thinks her father's still alive giving performances before cheering crowds. You really cared for her, didn't you?
SPOCK: Ready to leave Benecia orbit, Captain.
KIRK: Stand by, Mister Leslie. All channels cleared, Uhura?
UHURA: All channels clear, sir.
KIRK: Whenever you're ready, Mister Leslie.
LESLIE: Leaving orbit, sir.
MCCOY: You're not going to answer my question, are you?
KIRK: Ahead warp factor one, Mister Leslie.
MCCOY: That's an answer.
Deela in "Wink of an Eye" and Miramanee in "The Paradise Syndrome" are the two certain times he did, that I can think of. There are times when it was implied, like "Bread and Circuses," where I do not think it happened. He sent Drusilla away for ethical reasons.
I agree with you on Drusilla, and there were severe extenuating circumstances with Miramanee. I do have to add Odona from "The Mark of Gideon," though. The implication that Kirk slept with her is virtually inescapable.
That's a whopping three women in three seasons - and all three, incidentally, in S3. Some of the "Kirk as womanizer" nonsense comes from people with fourth-hand knowledge of Kirk's
past loves - Areel Shaw, Janet Wallace, the "little blonde lab technician," Ruth, and Janice Lester. Rayna, Shahna, and Edith Keeler also feed the beast. But none of that really proves anything.
As for Janeway dismissing Kirk, which I assume comes from "Flashback," I don't interpret that scene quite the same way either. Here it is:
JANEWAY: It was a very different time, Mister Kim. Captain Sulu, Captain Kirk, Dr. McCoy. They all belonged to a different breed of Starfleet officer. Imagine the era they lived in. The Alpha Quadrant still largely unexplored. Humanity on verge of war with Klingons. Romulans hiding behind every nebula. Even the technology we take for granted was still in its early stages. No plasma weapons, no multiphasic shields. Their ships were half as fast.
KIM: No replicators, no holodecks. You know, ever since I took Starfleet history at the academy, I always wondered what it would be like to live in those days.
JANEWAY: Space must have seemed a whole lot bigger back then. It's not surprising they had to bend the rules a little. They were a little slower to invoke the Prime Directive, and a little quicker to pull their phasers. Of course, the whole bunch of them would be booted out of Starfleet today. But I have to admit, I would have loved to ride shotgun at least once with a group of officers like that.
I think that Kate Mulgrew's acting in the scene conveys that she's teasing Harry - her greenest and most impressionable officer - with the over-the-top claim about "booted out of Starfleet." She was also probably trying to send a message to Harry that he should be careful until he reaches a position of more influence. But no Starfleet captain in Janeway's era would dismiss Kirk & Co., the literal saviors of Earth more than once, so cavalierly. And indeed, as a character Janeway displays Kirk's influence far more than Picard, Sisko, or Archer. I suspect that Braga, who wrote "Flashback," was indeed showing us a scene where Janeway told her junior bridge officer something in code but didn't really mean it, or he was just winking at the audience. The Kirk/Janeway parallels in the series are unmistakable, and it's not like Braga was a minor figure before or after "Flashback." The simplest answer may be that this was the 44th of 172 episodes, and Janeway's very Kirk-like tendencies evolved later.
Far sillier to me was the throwaway hallway dialogue by Dax and Sisko in DS9's "Trials and Tribble-ations," where Sisko - who deeply admired Kirk - comments that he was a ladies' man and Dax rebuts with the tired trope of being more romantically interested in Spock. Yawn.