I think the problem the Doctor has with Jack Harkness is he doesn’t see how Jack has changed and he doesn’t think Jack needs him like his other companions need him.
1) Obviously the Doctor and Rose met Jack when he’s more irresponsible and more like John Hart. Jack spent the majority of the first two episodes trying to convince the Doctor what was happening wasn’t his fault. That, trying to con them, and “You got to set an alarm for Volcano Day. [laughs]”
Of course the Doctor sets everything right, and Jack takes responsibility for what he done by getting rid of the bomb for them.
Jack Harkness thought for several moments he was legitimately going to die. He never considered the strange man he met would save him, so he drinks a martini and says goodbye to the only company he has (a computer)
When he mentions how the Doctor’s ship is bigger on the inside the Doctor replies “You better be.” Jack enters the TARDIS needing to prove himself.
I believe this first impression sticks with the Doctor forever. He views Jack as some sex-driven, conning, and selfish man and for awhile it might be true, but Jack changes.
2) Jack is very different to any other companion we’ve seen because of his abilities with technology and ability to take care of himself and others. We see him doing repairs on the TARDIS console in “Boomtown”, something we’ve only seen the Doctor do, and in the same episode when Jack specifically names the alien device the Doctor replies “Couldn’t have said it better myself.”
In “Bad Wolf” and “The Parting of Ways” we see Jack in his element. He escapes his game with no help (?), makes weapons, help the Doctor (specifically taking care of him when he thought Rose was dead), figured out Rose wasn’t dead, inspired people on the Gamestation to fight, and when confronted with certain death faced it bravely.
When the Doctor sent Rose home, he didn’t do the same for Jack. I think he thought he was useful. I think he thought Jack was better prepared and understood what would this battle entail. I think at his core the Doctor thought Jack was more expandable, even mortal, than Rose was. Maybe Jack thought he was too.
Jack was willing to die for the Doctor, and he did, and his faith was awarded by the Doctor leaving him behind.
The Doctor thinks the now immortal Jack is wrong. I can almost see his excuses for abandoning him now: he’s wrong and he’ll be okay anyway. He knew Jack could get off the Satellite since he’s so resourceful. He thought Jack would be okay emotionally because he’s Jack. Flirty, easy-going, tough as nails Jack Harkness.
I think at this point the Doctor becomes completely awful at understanding Jack’s emotions.
Jack changes over the course of ~130 years. He’s faced trauma after trauma. He’s lost so many people he’s loved. He’s so incredibly, achingly lonely. He’s trying so hard to be more like the Doctor, and he is, but not the parts he wants to be like.
And you’d think the Doctor would notice. In Torchwood season 2 episode 10, a little old lady sees Jack’s eyes are “older than his face” and that means he “doesn’t belong anywhere”. You’d think the Doctor would see this too, since the same is true for him, but I think the Doctor doesn’t. I think the Doctor sees Jack as the same useful although scheming former Time Agent. But now wrong, but now working for the institute he abhors.
And he uses Jack. He uses him over and over and Jack dies more in those episodes he was with the Doctor and Martha than all of Torchwood season 1. Jack let’s himself be used. And there’s nothing the Doctor can do about his immortality, or his slow aging.
Jack Harkness tries his best to be good. Sometimes he fails, and sometimes he fails spectacularly but he wants to protect as many people as he can. In Torchwood we see him as a deeply traumatized asshole. He carries responsibility in a way we’ve never seen him do before. We see him carry the responsibility of Torchwood, of his team’s fate, of the world’s safety. We see him so deeply care about his lovers and friends, and we see him so easily take blame onto himself. We don’t see that in 9’s season. After meeting the Doctor and Rose Jack opened his heart, and it’s been broken again and again, but he still tries. He’s a human. He’s a 51st century human and an immortal one at that, but he’s a human being that can be incredible and incredibly dumb.
I think the Doctor forgets that. Jack is wrong, Jack is thick skinned, Jack is easy going. Jack let’s him use his immortality.
The Doctor still sees Jack as that man he met during the London Blitz. He doesn’t see Jack for who he is now.
The only character devolpment he sees in Jack is his “wrongness”.