The game pretty much sets up the Brotherhood of Steel on the East Coast to be smacked down by a Good Karma player.
Epic, crushing defeat.
Likewise, the one on the West Coast is reduced to a shell of its former self.
I think a lot of people saw Bethesda as friendly to the BOS but they weren't. Not really.
The game kind of goes a long way to making you sympathize with Synths and ghouls.
You can have a huge number of them as Companions as well as Settlers.
They make it clear the BoS are planning to eliminate both.
Likewise, halfway through the BoS quest, you're asked to kill your a close friend in the Brotherhood for a ridiculous reason. I think the game sets the BoS up to be a dark gray faction which is probably not the canon ending.
Bethesda wouldn't want a Brotherhood Empire screwing up the East Coast for future games anyway.
There is also the debate about the MWBoS and what has become of them. They have been their own faction of a hundred years and have forgotten their history. I can't imagine they would give a crap about Arthur's authority.
As pointed out the West Coast Brotherhood might already be gone. New Vegas gives the strong impression that the NCR is clearly winning the war and Fallout 4 gives the impression that the Brotherhood will be staying on the East.
Why would they have forgotten their history?
They started out as the survivors of the ill fated air convoy mission. So not that many actual Brotherhood members to start with. They functioned by taking in large numbers of advlt wastelands. They would give them basic weapons and armour and pointed them at the enemy. So very little time for indoctrination and history lessons. They expanded rapidly, fighting from outside of Chicago to Colorado in a short time. Many raw recruits would get battlefield promotions to higher status in the groups. More wastelanders get taken (forced in) to the brotherhood to replace the loses.
So you have basically new Brotherhood members who didn't go through a life time of indoctrination now teaching raw advlts the was of the Brotherhood. Repeat that over and over again for several decades and you would have very few if any one left who would know about the original Brotherhood of Steel.
Just look at Arthur's people in 2287. They take in new people but it looks to be almost exclusively children or young advlts and they go through years of indoctrination and have a groups of people who dedicate themselves to history and yet they have bad records.
In New Vegas Caesar points out that Brotherhood captured in his eastern territory don't know of their own founder, Roger Maxson. The only Brotherhood that could be is the MWBoS in Colorado Springs, Colorado. That is where they ended up at at the end of Fallout Tactics and Caesar's Legion by the time of New Vegas started expanding into Denver Colorado.
Still, it's entirely possible they've indoctrinated all of them and their children.
Danse is the most Brotherhood Brotherhood man who ever lived.
And he's a convert.
The MWBoS didn't take in children. They took full grown advlts. There was no time for proper indoctrination. They were at war and that war would have and did take the lives of many of the original airship survivors. Thus as time went on there wasn't enough people and time for a proper indoctrination process. Fallout Tactics even showed that they had people join just to steal from them. To keep order the MWBoS became very brutal. Going so far as having deathcamps and using crucifixion.
This was the MWBoS' https://www.youtube.com/watch?v=I4acAV46vvQ
Saying that the RR is the only option for the good karma vault dweller is misleading. The RR are just as screwed up as every other faction. They kill synths so that new synths can hide elsewhere by wiping their memories. They abduct synths who don't agree with them and then kill them, too.
Yeah, the Minutemen seem the best option but I still think the BoS is getting the airship destroyed and Maxson killed.