Every time these debates come up I always ask why the brotherhood can't change after centuries. Original Fallout fans always scream nostalgia back as if it's a real answer, it's not.
However lets look at the reasons for the change:
This is years after F2, which is enough time for changes in leadership.
The Brotherhood in F2 was dwindling away and needed the chosen one to fight their battles for them.
The Brotherhood in F3 went to the Pitt where they carried out their usual mission while watching people suffering unlike anything in the first two fallouts.
This was without any other elders to remind them to stick to the basic tennants of the code.
The Brotherhood in F3 got pinned down and immediately started losing knights to the mutants.
Without any back up and with the mutants constantly dragging people off to recouperate their losses the Brotherhood would lose through attrition.
Helping wastelanders survive and getting fresh initiates went hand in hand with not having every wastelander become another mutant.
Even with the situation many brotherhood still chose to just hoard tech and left with alot of good technology further weakening the Brotherhood.
Now for the reasons why the brotherhood should remain the same:
Uhh... um, hmm... It's different so I hate it.
The original Fallout fans at this point strawman their way to arguments such as having the brotherhood grab tech and run which doesn't justify trekking the country at all and just not having the brotherhood which was never going to happen since like it or not, they are the face of Fallout.
I would have personally preferred a ghoul army fighting the mutants since it would make more sense through the lore but lore doesn't sell games marketing does. Black Isle couldn't market their way out of a paperbag which is why the series collapsed and why the original fans are now a vocal minority against a horde of F3 fans.
My answer is simply that the Brotherhood of Steel does not really have to changes at all at their standpoint of thing. Even at the time of New Vegas, they are still regarded as one of the tops in technology, knowledge, and weaponry. If there any changes in faction is needed, the Dev might as well make a new faction in correspond to their ideal rather than just pick out an old faction and make them "good" guys or "bad" guys whether the plots needed. At best, I can see only some adaptation of the BOS with the influence outside of their ideals, but still retains their way of live. At worst, they goes Steel Plague if push to the boundary of near wipe of the NCR. That would be an interesting route.
As for Fallout 2, the BOS just set up outpost all over the place in observation of the Enclave. The quest it self is optional and not needed but at the same time, it is pretty much the "Glow" mission of Fallout 1. Assuming that the Chosen One is really the descended of the fame Vault Dweller, they just sent the Chosen One to steal the plans rather than just do it themselves. More of an "efficiency" of the BOS part rather than needing help or that they are "dwindling". Beside, about two other factions within San Francisco wanted the said plans, so why not.
And as for the Lyon's Pride position, they might as well do what the Midwestern BOS did and charges the resident for supplies and other means if the Lyon's Pride plans to stay there. They cannot do all that fighting with the Mutant with all the "goodness" and free "protection" in the Elder Lyon's view of things. I am even surprises Lyon's Pride still exist in DC for 20 years.
As for ya "Uhh... um, hmm... It's different so I hate it." along with the next sentence is most likely the fact that such tempering of the Lyon's Pride is not needed and the BOS could in fact retain they overall looks in regards in their stay with DC, to again, something along the line with how Midwestern BOS deal with their stay if its that extreme. The way Lyon's Pride handle, they can't just keep fighting for free or in the worst case they are in right now, becoming Water Merchant minus the Merchant part. Sooner or later, giving out free water would likely cause an unfixiable damage to their own organization in term of men, morale, and already broken objective if they keep giving out free water.
Oh, and the Black Isle Studios were laid off before Interplay bankrupt themselves. Interplay wanted to make Fallout: Brotherhood of Steel 2 but needed resource to make it work. They did this by laiding off the Black Isle, which by that time were already almost done with Van Buren, by 95% done. Interplay put in resource for Fallout: Brotherhood of Steel 2 but is still short on resource which (along with other things that was going wrong with Interplay) result in the bankruptcy of Interplay. By than the Black Isle Studios join up/form Obsidian Entertainment.
That is essentailly the death of that chapter then, because they are no more. they are once more the main BoS, not the Mojave chapter.
Mojave BOS or the Western BOS? Mojave would just get out of there and reconnect with the Western BOS. In Western case, they are pretty much fighting the NCR and still giving them a hard time. The fact that they are still fighting and was able to sent an expeditionary force at the same time pretty say that they are still able to last a good amount of decades before being completely wipe out. I would not be surprises if the West Coast BOS goes Steel Plague on us.
It's not Elder lyon's pride though, it is Sarah's. The Elder has no say in who is apart of the pride (although in extreme cases I guess he would) Sarah formed it hence why she 'Handpicks everyone' The Elder uses it as a tool just like the rangers of the NCR are used. The pride wasn't even thought up until Sarah was old enough to fight and prove her combat prowess.
It might as well be called the Lyon's Pride by than. Though the only other name I can come up with is Wannabe, but Lyon's Pride is a bit "nicer" save the fact that is what mention half the time I was there in the Citadel. And again, like I said before, they are all under the leadership if Lyon, whether its the old man or his daughter after the old man dies.