Yes Kirk is a player (not that there Is anything wrong with having multiple sex partners as long as all parties are fine with it). But some very key points to remember, outside of Beyond, the nuTrek Kirk is shown at an earlier stage of time, and has 20 yers of divergent experiences. But TOS Kirk isn't.exactly a shinning example of how to behave with women, let alone how to behave with people in the work place. And in those three years we see almost nothing showing a progress of learning, on this subject.Thing is, he goes through the same thing with Uhura in Trek09. From hitting on her, ogling her while she undresses and respecting(?) her in the end. You'd think his frat boy behavior would stop there, but for some reason 'Into Darkness' decided to dial it up by having Kirk wake up with two alien women in his bed and the film immediately cutting to him hitting on two more women on his way to meet Pike.
He will respect women he knows, but there was no indication in 'Into Darkness' that his constant hitting on women en mass is something he's ever going to stop doing. Beyond didn't feature any of that because the writers thankfully didn't put Kirk in that kind of position.
And yeah. I would have liked it if Carol stuck around. But sadly, if you're not Uhura, all women in NuTrek are one shots. Which is sad because that applies to Jaylah as well. At least Beyond remembered that Kirk has a living mother.
Now I have issues with every version of Trek, both from the lens of the immediate reaction and then to placing those shows, or films in the era that they are made. But there is a huge difference in that in TOS, we get 78 episodes with this person and we get to see how he (and everyone else) behaves, both their strengths and weaknesses. With nuTrek we got 2 brief moments in time (his childhood and bar fight leading to academy), and three missions. We get a tiny window into the characters. So every good or bad behavior is going to seem so much more significant than than any good or bad behavior.