I have watched every episode of every series--and more than once--and I guarantee that Kirk never used his charisma like that.
Janeway wasn't exactly standing there with tears running down her face and and wearing a massive pout. The point I'm trying to make here is that both Kirk and Janeway used their natural charisma as part of their leadership style. They both had close friendships with members of their crew and often had to ask them to do things they wouldn't do for others.
And the point I'm making is that she wasn't *asking* Chakotay to do *anything*. He already said he'd follow her orders, even though he disagreed with them.
She was emotionally using him, manipulating him, engaging in a pity party.
I behaved the same way she was behaving when I was 14 and wanted my father to let me do something he thought was a bad idea. It was a bad move with my father and it was a bad move for Janeway.
You know, interpreting facial expression and mood is a chancy thing. To me, Janeway isn't saying she's alone in a pouty or manipulative way. As you point out, the argument is over. He is going to follow her orders, so, really, there is nothing left to say to him as her first officer.
To me, she is talking to him more as an equal, letting the formality of captain/first officer drop momentarily, because what she murmurs echoes what they had said to each other privately, as friends, earlier in the day. She has come up against the reality of her situation in stark relief--her crew can promise that she won't be alone, even in a spirit of friendship, but, ultimately, the captain IS alone. She bears the onus of command on her shoulders and has to do what she believes is the best, even if her crew (and her friend) disagree.
I think the tears in her eyes come from the sorrow she feels about what has happened between them more than pity for herself. Their attempt to bridge the chasm between captain and crew has failed. The comment and the tears in her eyes show us that she has decided against his advice at a high price and that she deeply regrets having to do it.
In the use of her personality and charisma, Janeway is very much like Kirk. I, too, have seen all of the TOS episodes and movies, and it is obvious that Kirk knowingly flirts and/or manipulates people whenever it serves his interest--and especially when he is confronted with a female. It is such a chiche, that, in "The Undiscovered Country," when he befriends a female on Rura Penthe and the woman kisses him, Bones comments, "What is it with you, Jim?" I still chuckle when I remember that scene.