There is something undeniably weird about the new Kirk that we're seeing in Strange New Worlds. He doesn't yet "feel" intuitively like Kirk to me, especially in the rom-com episode. But I do think his writing and, to a lesser extent, his performance show that the writers are thinking deeply about the character and what people have been missing about him. In a sense, SNW may be trying to counteract the phenomenon of Kirk drift, where pop culture stereotypes about the character's impulsive, womanizing ways makes it impossible to understand the person we actually see on screen.
What the first season finale shows us is a Kirk who is by the book, yet decisive and sure of himself. He does not disobey Pike, but he is not afraid to tell him he's wrong -- not based on gut feelings, but based on a sound tactical analysis that proves to be right. Compared to Picard, Kirk -- especially the movie Kirk -- may seem brash and prone to violate the rules, but TOS consistently shows us a captain who respects authority but is willing to push it up to the very limit to protect his crew and achieve his goals. It's interesting that the episode picks up on this aspect of the character as the one that creates an instant bond with Spock. It's not his emotional nature or his instincts or whatever else, it's his respectful yet firm leadership style -- a sharp contrast to Pike's tendency to leave his subordinates to their own devices.
In the romcom episode, the message is a little garbled by the fact that this is an alternate timeline Kirk, but I think it highlights the fact that (a) Kirk is not a compulsive womanizer by any means and (b) Kirk bonds sincerely with women who feel isolated by leadership or other burdens -- not in a predatory way, but in an empathetic way. In contrast to Chris Pine's layabout troublemaker who is constantly getting laid (at least in the first film), the Kirk from TOS is basically a lonely nerd. A charismatic one, to be sure, but still a lonely nerd. Even well into his second command, he's haunted by the guy who bullied him at the Academy! He is, if anything, sexually thwarted by his sense of duty and his "marriage" to the ship. Hence when he meets a woman with a similar predicament, they are drawn to each other. Everyone has a type! It's just a sad coincidence that he wound up meeting someone of his type virtually every episode in season 3.
I don't think it's perfectly executed, at least in the pairing with La'an, but I do like that they're trying to refresh our perspective on the character and that they're doing it in a way that reminds us of all the traits from TOS that the pop culture parody of "Captain Kirk" leaves out. But what do you think?
Not exactly what you're getting at here, but the first scene in the last episode, where we see Kirk on the UEF Enterprise, as La'An enters the bridge, and he turns to her, and talks, that is absolutely channeling TOS Kirk in virtually every aspect.
And then, outside of that, In both of the episodes we've seen him in, we've explored aspects of Kirk that are both the same as the Kirk we know, as well as very different from the Kirk that we know.