It is most assuredly not the whole point.

To be frank, some of the discussion of AGoT!Jon seems like it is stuck in the chapter where he bullies the other recruits and is generally bemoaning his place at the Wall, and simultaneously fundamentally misunderstands the problem Jon’s behavior showed: Jon’s flaw was not that kept whining that he was a bastard or however he gets described, it’s that he could only see his own pain and his own problems and that he took it out on the others around him while implicitly expecting sympathy and recognition for what he was going through without caring to look farther than his own troubles; it’s that he could not recognize that his own behavior–that of an entitled brat–was why he was met with such a chilly reception; and yes, it’s that he could not recognize his own privilege and only saw his disadvantages. But Jon was disadvantaged. Erasing that to only focus on his privilege not only undercuts a major part in Jon’s story and identity arc, but also strikes me as rather unsympathetic to a boy of fourteen. Jon definitely enjoyed perks that many other bastards and most recruits on the Wall did not, but that does not change the fact that he grew up a bastard in a household whose chatelaine was invested in putting him in his place and asserting that Winterfell was most assuredly not it. This is a major point of contention in fandom, but I personally classify Catelyn’s treatment of Jon as emotional abuse, which should not be ignored when we’re talking about him. That’s why some of the discourse around
this part of Jon’s story rubs me the wrong way because it smells too
much of an expectation for Jon to just get over it already and grow up!
He is whiny. He is broody. He is too caught up in his own pain, etc etc
etc. He is freaking fourteen, and there is no set time limit for someone
to get over trauma or they’d cease to be sympathetic or relatable.
That’s a disturbing train of thought, and unhealthy expectation to put
on a kid, especially one who already had to contend with unfair
expectations being placed on him, as demonstrated in the conversation
between Catelyn, Ned and Maester Luwin.

That, however, does not mean that Jon should not be held accountable for his actions at Castle Black, especially since his initial deeds were not exactly acts of solidarity to his fellow downtrodden. The narrative does not expect us to be on Jon’s side when he was lording his physicality over Tyrion, or when he was proclaiming his superiority to the other recruits or when he was expecting favorable treatment in demanding he accompany Benjen in his ranging. Neither of these moments were exactly a ringing character endorsement for Jon, and the narrative treats them as they are: shameful, mean and entitled behavior in which Jon was using his privilege, whether his physical ability or his castle-bred training, to bully others. He was being Theon Greyjoy, plain and simple. No, really. Jon’s thought process to explain why the other recruits hated him is too reminiscent of Theon’s own thoughts about why Jon hated him. That’s why Jon earned
Donal Noye’s verbal smack-down to shake him out of that behavior and that narrow-minded view, and why Benjen had to tell him point-blank that he hadn’t earned what he was demanding. Jon needed those wake-up calls.

But while the narrative gives Jon’s behavior the reaction it deserved, it also does not neglect to remind us that Jon is fourteen years old. He is a kid, and kids do not typically see past their own troubles. His behavior was unacceptable, absolutely; it was petty and entitled and childish, but Jon was a child, and one who was dealing with feelings of abandonment and displacement, desperate to prove himself and find a place where he was not demeaned. The thing about this period of Jon’s story is that it represents the learning curve he had to go through in order to be the most suitable choice for interacting with the free folk later on. The point about these AGoT chapters is that Jon learned. It is not easy to recognize your privilege, neither is it easy to have your worldview challenged and to accept whatever change that challenge brings. But Jon managed both. He listened when he was rebuked for his behavior. He felt shame and unease when his misconduct was pointed out to him. He changed his attitude and went from looking down on his fellow recruits to helping them and even protecting them in the case of Sam. When he grew angry at his appointment to the stewards, Daeron and Sam’s words made him ashamed again for his tantrum. That’s not an easy thing to do; grown adults have trouble owning up to their mistakes or accepting challenge to their worldview, but Jon did both. That ability to revise his stance on things and to face his mistakes, his willingness to learn and desire to be better marks a very important part of Jon’s character, and is frankly one of the main reasons I love his character so much.

Those attributes are what propelled Jon to recognize that their society’s definition of masculinity was deeply flawed, and the stance that worth is ascribed or denied to a person simply based on their martial ability was faulty, which is why he saw the value of someone like Sam, a self-proclaimed craven based on Westerosi code of masculinity, but who is actually extremely brave and extremely valuable to the fight against the Others even if his bravery and his value do not lie in performative masculinity or on the field of battle. Or someone like Satin who does not fit masculinity standards at all with his traditionally feminine self-presentation and his profession that attracted the ire of other watchmen even more than literal rapists and murders simply because it was feminine-coded, but who is also loyal and steady and able as shown by him holding his own as a new recruit even in the face of the disarming attack from the free folk. Those attributes are what make Jon one of the most radical thinkers in the entire series. He was someone capable of facing his own prejudices and preconceived notions and conquering them. That’s how he was able to look beyond the bigoted view of the free folk and see them as simply human and equally entitled to protection and safety as much as anyone living south of the Wall. Jon, through a learning curve that started in AGoT, demonstrated how he was the perfect leader for the War for the Dawn because he proved that he was capable of looking beyond any and everything to see the common feature between warring sides–their humanity– and recognizing that this was the thing that mattered most in their fight, and that this shared humanity is what he was fighting for in the first place. That’s how he understood that holding onto old constructs about the evilness of the free folk was not acceptable anymore. He had the courage and the prudence to think about what his oath truly meant. But that ability did not come from nowhere. It was a journey that started with a privileged bastard taking his frustration out on his fellow recruits, to a lone ranger living and building relationships with the people he was taught were enemies, to a Lord Commander insistent on using his power and his position to do good. But the latter could not have been without the former, could not have been without the tutelage of Donal Noye, Benjen Stark, Jeor Mormont, Maester Aemon, Qhorin Halfhand, Mance Rayder, Tormund Giantsbane, Stannis Baratheon, and most importantly, Ned Stark himself who taught Jon the value of human life and who treated his lessers as human beings.