Where is democracy in any of the Fallout games? the NCR is the closest thing to it and they are presented as constantly struggling for resources and already falling into corruption. Pretty much all of the concepts you call out in that first line are things you get under rule of law and in Fallout that stretches are far as your aim is steady. The ones settitng those laws are usually the most violent, manipulative or the ones who were born into those roles.
They were founded by a group that defected from the US military/government after seeing how corrupt it was, they saw themselves vindicated when the war happened and believe that being raised in their group instills a proper sense of morals regarding the use of technology, it also fuels their distrust of outsiders which we are already seeing is a changing point of view. so they are wrong when viewed from a near omnipotent outsider, doesnt mean they dont believe it.
Honestly i like that they arent cut and dry democratic and egalitarian goods guys. the only ones who are close to that are the Minute Men and the game shows you where that got them, when faced with a serious threat such as the Gunners most of the Minute Men decided not to risk it and let Quincy get gutted.
The BoS is a path towards stability at what will likely be a horrific cost. That still makes them a lesser evil than most of the groups they fight against.