The penultimate episode of Billy the Kid embodies a distinct sense of being a side quest, albeit not entirely unfair given its intricate intertwining with long-standing character arcs and leading up to a notable, albeit ahistorical, climax. Season 3 has veered slightly off-course since it was revealed that Billy had miraculously survived his shooting by Pat Garrett, and Episode 7, "The Last Buffalo," marks the manifestation of an epilogue to the real story.

It's a captivating concept to delve into Pat's post-shooting mental state. He dedicated his life to hunting down Billy, believing he had succeeded. But the disappearance of Billy's body and the Mexicans' near-worship of him left lingering doubts that were picked up on. He was relieved of his duties as Sheriff of Lincoln County and continued to fall further into disrepute, working menial odd jobs and succumbing to alcoholism. He wrote a book that no one cared about, one that only perpetuated the legend of Billy, making a mockery of Pat's own exploits.
This could have been a terrible fate for Pat. But Billy is determined on revenge, so "The Last Buffalo" revolves around Billy and Jesse pursuing him for a final confrontation, using the book he wrote and news reports, many stemming from his own ego, to trail him across the country. It's a unique and clever structure, but it doesn't make for an especially captivating episode, with much of it spent watching Billy and Jesse lounging under trees and laughing at Pat's latest public disgrace.
Catron is not handling the ambiguity around Billy's demise well either, and his refusal to believe that Billy is dead—tightening his home security, increasing the reward for his apprehension, and publicly stating that Pat didn't kill him as he claimed—hastens Pat's decline. But Catron's lifetime of corruption has left him to face his fear alone. Even Emily, now armed with knowledge of Catron's involvement in Edgar's "suicide," couldn't care less. Catron's paranoia is amusing in this episode, as though he spends all his time tracking down Garrett despite Billy ultimately intending to target him.
Garrett has reached his lowest point, ensuring his inevitable downfall. As a blacksmith's assistant, shoeing horses, he succumbs to the psychotic 16-year-old daughter of the blacksmith, who manipulates him into a situation where she essentially forces herself on him and then tells her father that the situation happened in reverse, for no reason other than her own amusement. Garrett is branded by the blacksmith, and the story becomes news; Billy and Jesse sit laughing about it under a tree. From there, they track Garrett down at his latest job, guarding prisoners on their way to jail. In the middle of nowhere, they ambush the prison convoy, and Billy presents himself to Garrett like a specter emerging from the depths of his own mind.
Knowing that Billy is still alive would have been fate enough for Garrett; previously, it was the uncertainty that haunted him. Seeing Billy in the flesh unravels every story Garrett had told himself about his own heroism. He was already ruined; this insult is a final indignity. But it's not enough for Billy, who does him the courtesy of allowing him to draw first in a final duel. Billy wins easily, executing Garrett with a shot through the heart, exactly the same fate that the former sheriff had attempted to visit on him.
As you may wonder what could possibly come next after "Billy the Kid" Season 3, Episode 7, Jesse repeatedly tells Billy that he's wasting the second chance, the "rebirth" that he has been given. He could be spending his time living blissfully with Dulcinea and their child. But his thirst for revenge consumes him, and it isn't just Garrett whom he wants to pay. There's Catron too, and the entire Santa Fe Ring, which he believes is his moral responsibility to shut down. While Jesse was willing to help Billy hunt Garrett, he's not willing to take on the seemingly impossible task of battling against institutional corruption. That, at least, seems like a fight that even Billy can't win, and he'll be riding into it alone. Surely the show wouldn't spare Billy from his historical fate only to kill him off in a different way, would it?