I have not seen anyone saying Lyons or anyone after Lyons will be or should be a Tyrant.
A lot of people clearly don't like what Lyons is doing....he and his men selflessly try to protect the people of the CW, solely because it's the right thing to do, and they scream "White Knight" and say they want more "shades of grey". Then the NCR comes along with those "shades of grey" and they scream "they're corrupt and don't care about the people of the Mojave, give us Caesar!". So I think they indeed would prefer Lyons to be a Tyrant, or at least a Machiavellian b*stard feigning concern about the people so he can manipulate them to his own ends.
I myself am not against the idea of the CWBoS forming a government, IE "The Capital." I have just been posting questions and factors that could effect any future expantion/growth of this "Capital." Such as the CWBoS are "White Knights" and they will not become Tyrants.
I think Lyons is more likely to nudge Rivet City, Megaton, Arefu, and the Commons...along with Tenpenny Tower if they are willing...into forming some kind of unified government on thier own, with guidance from him and the LW, then throw the BoS's weight behind it. He could also decide that this Government's Secretary of War (or whatever title they use) is the legitimate successor of the US Secretary of War and chuck out the Codex entirely and return the BoS to it's roots, as the Army. The BoS would revert to civilian control as the national Army, Lyons would retire as Elder and would probably be easily elected President, and become the new nation's George Washington, setting by example the political norms of the new nation.
So what would happen if they come across another peaceful faction but that faction does not want to help or join the "Capital." As you put it they would not try to annex it or take it by force. This would slow their growth. Such as Rivet City and Megaton.
What would happen if the Outcast can't be negotiated with but are actively working against the "Capitals" goals?
People that don't want to join would probably be left alone as long as they weren't doing things that were repellent, like slaving, cannibalism, or being actively hostile. Once Lyons is demonstrated as having being right all along, I wouldn't be surprised if at least some Outcasts petition for pardon and reinstatement, which Lyons would probably grant. With the Enclave and Mutants dealt with, Lyons would certainly launch an all-out war on the Raiders, Talon Company and Paradise Falls.
What about Ashure? What if he does end up creating a large army?
He already has one, but neither he or Lyons can afford to make war on each other...Pittsburgh is 250 miles from Washington. Once Lyons wipes Paradise Falls off the map, Ashur will have to look elsewhere for workers. By the time the Capital and the Pitt are large enough to fight, Sandra will probably have a cure for TDC and Ashur will have the luxury of using carrots rather than sticks to keep his factories running. Lyons could accept the existence of Ashur as a King so long as he isn't shanghaiing people to work for him. For Ashur's part he would probably be delighted to learn that Lyons was attempting a kindler, gentler version of what his is trying to do.
What happens if the project p breaks?
Sounds like the main Quest for a new Fallout.....finding another GECK.