Lyon's chapter did not give the original BoS style of life, he just tweaked it a bit.
When he did go to DC, he passed through the Pitt and killed the slavers and everyone who was infected.
The sight of the pitt , was a waking up call for Lyon.
While still looking for tech, Lyon took the task to actually improve a bit the life of the people in DC.(Project purity for ex)
BoS was pictured as knights in shinning armor in FO3, players that actualy know more about the lore, knew that isn't actualy true.
The BoS stick mostly to their own, makes only contact with people if they can get tech, get food or kill. If you don't have anything to offer, they won't lift a finger...