Pickett was a Virginian, in command of Virginians, and Virginia was a state that was kind of full of itself. It also helped that, of course, the rebel army retreated into Virginia as well. This meant that the first recounting real of what had happened was too Virginians, and thus the setting the stage for how the charge would be presented in conventional wisdom. According to Pickett's Virginians, as they had been the furthest progress, the charge could surely not be blamed on them. They had done their damndest at the right wing, and it was those further over who failed to adequately support them, and doom the action to failure. No matter that the worst performing brigade under Pettigrew’s command was, in fact, also Virginians, the narrative in Virginia quickly arose that the charge had been a glorious one for their own, and it had been the performance of the other states' units, principally North Carolina that ensured its failure.

North Carolinians voiced their defense, pointing to Brockenbrough’s Virginians as the real cowards who had turned and run, not their own men, but they lacked the same strength of the press as in Virginia, and it didn't really matter, since in the Army of Northern Virginia, they were quite outnumbered, and found themselves receiving the cold shoulder from Virginian comrades showing what they thought of them. the issue became so dire that it was tied to rising desertion rates by North Carolinians during the summer, and several resolutions back home passed calling for an end to the war. By 1864, the issue had mostly fallen to the wayside, as the war continued and more immediate matters took precedence, but the argument was far from over, and would resume after the war as veterans from both states continued to fight with the pen over whose fault was the defeat.

There is a bit more to be said about the argument itself as it continued on for some time after the war, but for the question at hand what is more important is two fold, first that the argument was one that happened, and second is that the early, dominant narrative was one which gave the glory to Pickett's Virginians and lumped the failure elsewhere. Assigning Pickett's name to the charge was not reflective of placing the failure on his shoulders, but in fact assigning him what honor could still be grabbed from the clutches of overwhelming defeat. With so much focus in the Virginia press on Pickett's accomplishments, he became indelibly linked with it in popular memory. For Northern veterans, who might have known the specific unit across from them, but less sense of the higher command structure, there was generally uncritical acceptance of the name in their own vernacular, only helping further solidify it. Even those who might have realized he was only one of several Divisional commanders often accepted the broader narrative, repeating in their own post-war regimental histories how "North Carolinians turned and fled in terror, Pickett's men alone remained".

After the war, post-facto justifications were made, often by Virginians, to keep the name attached. The rising sentiments of the Lost Cause, which celebrated the honor of defeat and asserted a pride of Southern manhood that fought gloriously against insurmountable odds only further added to the glory of the charge in the minds of many, and thus to the man who had led it. It had become that great 'what if' point, the moment of dreams for every proverbial 14 year old southern boy, as Faulkner famously intoned. Pickett's name had been attached somewhat organically, but now other reasons too ought to be added on, as in many veterans circles outside of Virginia, there was something of a level of displeasure, if not outright offense, at the degree to which the Confederate hagiography seemed to be so centered on Virginians to the exclusion of others. As such we see arguments from figures like, Fitzhugh Lee, a former rebel officer, writing around 1894 that:

The operation of a detached force generally takes the name of the commanding officer. Pickett was the senior officer in rank [...] The appellation refers to the whole body of nine brigades, and should not be limited to the three Virginia brigades which belonged to Pickett's division.

Lee's words also reflect the trend that came to be seen by the end of the century, which looked to move past the infighting of blame, while retaining the place of honor for the Virginian Pickett. A similar sentiment can be found in the words of Robert Douthat, a Virginia veteran writing a history of the charge in 1905, who by then described the charge of one of unified honor, where there was "certainly glory enough in the Charge for all."

Of course though, if one person was truly unhappy with that development, it was Pickett himself. His career suffered greatly in the wake, likely exacerbated by depression over the fate of his command. He would lead several inconsequential actions over the next year until finally directed by Lee to command at Five Forks on 1 April, 1865, which he thoroughly bungled, helping speed the way to Appomattox. While his plaintive reply to Lee on July 3rd that "I have no division" is not well attested to, John Mosby relates after the war a discussion with Pickett where he blamed Lee for "massacring" his Division.

It didn't help that some in his own command held it against him that he had stayed to the rear rather than lead from the front, a charge that was of course immediately picked up and repeated by detractors from other states. A mostly unfair criticism to level at a Divisional commander, it nevertheless was a further string to the man's pride, and a fight that would continue in Confederate veterans circles for several decades until finally mostly settling in the 1880s. Whatever the intentions behind it, to have his name attached to one of the most bitter episodes of his career was anything put honorific.

Sources

Gordon, Lesley J. "'Let the People See the Old Life as It Was': LaSalle Corbell Pickett and the Myth of the Lost Cause" in The Myth of the Lost Cause and Civil War History. eds. Gary W. Gallagher and Alan T. Nolan. Indiana University Press, 2000. pp 170

Hess, Earl J.. Pickett's Charge--The Last Attack at Gettysburg. UNC Press Books, 2010.

Reardon, Carol. Pickett's Charge in History and Memory. UNC Press Books, 2003.