It's simple....not the process mind you, but Caesar's mindset.
Caesar is playing to win. Caesar believes in a selfless society free of corruption where citizens are capable of putting the greater good above their own well-being. He's disgusted by corrupt brahmin barons and self-interested casino owners alike, and he wants to change that. How do you beat greed? His strategy: indoctrination. Little kids are susceptible to all kinds of ideals. You teach someone young that greed is bad, they'll listen. This is why Caesar stresses the importance of integrating children into his Legion and why he wants women pregnant around the clock. This is also why women do not serve in the Legion: not because he believes women inferior, but because pregnant women are inferior warriors, and he does want pregnant women for the sake of having more children to indoctrinate into the Legion's ideals.
But Caesar recognizes that the NCR and other nations with opposing ideals are strong, and that if he's got any hope to beat them, he needs to give 110%. Thus, the Legion thusfar is built purely for war. War is what the Legion specializes in, and you better believe they're amazing at it. That unarmed Legionaires are capable of bringing down Super Mutants and Brotherhood of Steel paladins alike is a testament to their strength, their cunning and their general know-how of warfare and tactics.
All of this is done because Caesar wants absolute victory so he can wipe the slate clean, and then AFTERWARDS has absolute say in what society is like.
In short, the Legion post-victory would begin to look nothing like the Legion we see now. It's "ends justify the means," plain and simple. The Legion doesn't hang people up on crosses because they're dikes, they do it because they're playing to win and interested in the "greater good" of humanity, and they believe a Legion victory is more important than the suffering of one individual, because a Legion victory is, in his mind, capable of making humanity as a whole better off. Thus yes, that captured NCR soldier will suffer on a cross if it means scaring the pants off their enemies and helping to assure Legion Victory long-term.
How the change would come about, we don't know, and the game does not deny there would be some difficulties with this. For example many Legionaires missed the memo that women are not actually inferior and have become sixist, even if Caesar himself is not. It's also a matter of turning a bunch of dedicated legionaires who once knew nothing but combat into working-class citizens capable of farming and commerce, supporting the Legion in day-to-day tasks. Caesar himself will say that there are bad parts of both the Legion and NCR that will be weeded out post-victory, and I believe he's referring to the Legion's inability to produce anything themselves; thusfar, the Legion has only taken. The Legion themselves will admit they're better at taking and capturing strategic locations rather than holding them. The ending and Caesar's own logic implies that many of the citizens of Vegas would just....well, end up being citizens of the Legion. They'd continue doing their work, but now they serve the Legion. I'd also imagine casino owners and bar owners would be forced to shut down and told to go farm or something.
But yes, no one's denying there'd be difficulties making this transition, but unfortunately we don't know Caesar's whole plan on how to make it happen. We only know he fully recognizes the potential problems.