Lyons doesn't know how to lead. He is passionate about the goals he sets for himself and these blind him to ultimately more important issues. He always "hunts his white whale", first the Super Mutants then the Enclave, because fighting is all he understands. He showed no real understanding of politics or the importance of social infrastructures or government.
Seriously, nobody lives in the downtown DC area. It would have been easy to just contain the Super Mutants there and then move onto policing the outer wastes. There is almost no BoS presence there, as they instead choose to mess around hiding in their Pentagon hole, so real issues like slavery, raiders and the Super Mutants in Vault 87 are left unattended until you do all the work for them. As they have no knowledge of government or technology outside of weapons I can hardly see how they could help people make food for themselves, or other REAL ways of helping the Wasteland (and nice job Bethesda purifying the river water at the MOUTH of the river, because that will help) The only real help they would be good for are as caravan guards
The "Outcasts" had it right. They should have stuck to their original goal of collecting the huge wealth of tech in the DC area then returned to the West Coast. They don't know how to lead a people so they doom themselves in trying, especially as they don't have the numbers to police the entire Capital Wasteland. Back in the West Lyons could have told more people of the DC people's plight, and then REAL groups like the Followers of the Apocalypse could have gone to help. Seriously, how does breaking yourself away from the BoS's main group help at all? As Lyons was high up in the Brotherhood (unlike Veronica in New Vegas) he could have gained more members to join his more charitable cause, maybe even help change the Brotherhood modernise beyond their outdated values. But by becoming a renegade he loses all credibility.
Their actions are not noble. They are incompetent and ill-thought out. Even their giant robot got blown up. They have doomed themselves, and with them the hopeful people of the DC area who hoped for saviours from huge number of problems plaguing them, and instead they got a bunch of idiots wanting to play Arthur's Noble Knights in Shining Armour.
(Sorry, went a bit mental there. It was interesting thinking on this, though I won't vouch for all my points made being perfect, but the gist is there)