House states clearly that war with the NCR is not his goal, and it’s true.

Yes true, but if you side with House the outcome is he drives the NCR out for control of the Hoover Dam regardless.

Mr. House's Securitron army took control of Hoover Dam and the Strip, pushing both the Legion and the exhausted NCR out of New Vegas. Mr. House continued to run New Vegas his way, a despotic vision of pre-War glory. The streets were orderly, efficient, cold. New Vegas continued to be the sole place in the wasteland where fortunes were won and lost in the blink of an eye.

IRT why I think the legion collapses without Caesar:

Robert House: "By my calculations, his death will affect the shape of the battle for Hoover Dam minimally, if at all. The Legion's aggression will outlive Caesar. Indeed, they'll try to take the dam as a tribute to his memory. Given a year, they'd have him deified - but by then the Legion will be breaking down, riven by internal conflicts, a monster consuming itself. It's irrelevant. In the short term, the Legion is still monster enough that defeating it will make me look powerful indeed."

Joshua Graham: "I think only Caesar can lead the Legion. I've never met anyone who could take his place. I couldn't. I never had a mind for logistics. I don't know Lanius, but from what I've heard, he has no interest in leading anyone unless it's in battle. No. The Legion dies with Caesar. What follows now are just the last steps of a man who does not yet realize that he's walking dead."

Marcus: "{contempt} Caesar thinks he can change human nature. Most of the Legion is following Caesar, not Caesar's ideals. When he's gone, it'll crumble. Might not happen overnight. Might take a few decades. But it'll happen. Basic human nature - greed, ambition, jealousy - will see to it."