It's pretty flimsy (what I'm about to say, not what you've said), but he could be a town Marshall and maybe it's a matter of pursuing criminals that have committed crimes in Armadillo, and he can't act against criminals who commit crimes elsewhere.

Williamson's gang if I remember correctly didn't commit crimes in Armadillo to the same extent as Walton's gang and the Bollard Twins, and most of the time his missions involve pursuing criminals after they commit crimes in Armadillo.

Using your examples:

* They pursue Walton from Armadillo to Pleasance House, rather than seek him out there.

* The Pike's Basin gang had been rustling cattle from farms near Armadillo.

* The murders had been committed in and around Armadillo, and the trail of blood led the group to Ridgewood.

* Johnson himself was called out by Williamson's gang to get Bonnie from Tumbleweed.

* Even flimsier, but taking down Williamson's gang at Fort Mercer seemed to be a personal favour to John, not a decision he himself made.

The problem is that, he can't seek out criminals outside of town, even if they committed crimes in Armadillo, because that would be outside his jurisdiction.

While maybe pursuing a criminal (Walton) might be an exception to that and I can see Johnson going to Fort Mercer as a favor rather than an obligation, investigating the criminal activity in Pike's Basin and Ridgewood Farm would be the job for sheriffs and federal marshals, not town marshals.

Interesting observation. It would appear this inconsistency somewhat extends to Marshal Tom Davies from Red Dead Online as well. In the mission Highly Illegal And Highly Moral, Davies claims his jurisdiction as a marshal "ends at the city limits" (in regards to Van Horn). Although the previous mission in his circumstance features a different explanation pertaining to evidence, given both Johnson and Davies are stated to be members of the United States Marshals Service, in reality, their jurisdiction should be federal.

In my view, it seems the devs may have simply confused elected marshals as being members of the United States Marshals Service in these situations.

This is also a problem with the character Oswald Dunbar. The game can't decide whether he's a sheriff or chief of police. The characters call him a sheriff, but the games subtitles and objectives refer to him a chief.

Also, why is Dunbar, the chief of police of Blackwater, out in New Austin? Even if he was a sheriff, that wouldn't make any sense either, as sheriffs operate only on a county level, so his jurisdiction would only be in Great Plains.

In regards to Dunbar, though he is referred to as both titles in dialogue interchangeably, that can be chalked up to the term "sheriff" being used to refer to him as the head lawman in the area within the context of the conversations. Taking into account the specific circumstances of the aforementioned situation, details surrounding it could perhaps have been vague to most (save for Horley, Old Man Jones, and the players) as he may not have wanted others to catch wind of the mishap. The outlaw had also escaped from Blackwater, so it can sort of justify his presence there.

From some implications in the first game, however, it did seem as if New Austin was more a territory rather than a state. Of course, the second game does appear to confirm it is a state, despite it being undeveloped and notably lacking a populous settlement like Annesburg or Saint Denis. Additionally, the ties the area holds with government lie in Blackwater as well in both entries, so perhaps it is still meant to be considered a territory, in spite of it being referred to as a state.

Yeah, New Austin's status as a state was pretty shaky in between games. Even West Elizabeth was treated as the same state as New Austin, bounty wise, in the first RDR. Then in RDR 2 suddenly they are both different states. The jurisdiction between the states hasn't exactly been consistent. You can just chalk it up to the writers making a mistake or taking liberties with what the title marshal means.