Lyons BOS also shot Ferals, Super Mutants and Raiders on sight, they were just not very good at it, split into two factions (Lyons and Outcasts) and despite recruiting wastelanders were still in obvious decline, but for Liberty Prime and the Lone Wanderer they would have been an easy kill for the Enclave. Aside from giving out free water there is no a great difference between Lyons' and Maxson's vision for protecting the locals, beyond the fact that Maxson seems to know what he is doing, apart from the problem training Vertibird pilots.
The Brotherhood was never a democratic organisation in any of its divisions (West Coast, MidWestern or East Coast), Lyons was appointed Elder and sent east, Maxson is the guy who reunited the East Coast BOS, rebuilt their strength and turned them into the powerhouse of the East Coast.......most of the facist claims about Maxson's version of the BOS don't make a lot sense, since all of the BOS factions act the same way, even in Lyons time, the Brotherhood settled its problems through the barrel of a gun (or Energy weapon).