You can't just dismiss Edelgard's desire for forging peace with Almyra as a throwaway line. It's the conclusion of her whole paralogue! There is precious little room in those ending cards - they're barely longer than a tweet - so between eliminating TWSitD, reforming all of Fodlan, and personal information on her and her partner, there isn't room to reiterate a fact that they used her entire paralogue to establish. And even then, in the DLC she had to spend her entire A support with Constance presenting a TED talk to the audience of how her proposed reformations will work, because trying to change things invites so much more scrutiny and apparently the base game wasn't enough for everyone. Regardless, we get more of Edelgard's thoughts on foreign relations on record than we do with Dimitri.
I see no reason to believe that Claude is being anything less than honest with Edelgard during the post-Deirdru scene. He has no reason to lie about
the promise he makes to repay her in the future or about his ambitions for the future, especially since it isn't the only time he alludes to those ambitions. He has his dreams for the future, but he also seeks the power to make those dreams a reality. Which is why he's so drawn to Byleth to begin with and asking pointed questions about relics and whether the Sword of the Creator really can destroy an entire mountain - perhaps with a particular east Fodlan mountain fort in mind.
And he needs that power because there's no changing the church without first removing Rhea from power, and Rhea is the immovable object to Edelgard's unstoppable force. There is no clever scheme that is going to budge that immovable object, which is why Claude is seeking secrets and power. At least until Silver Snow asserts itself and insists Edelgard went too far, despite Claude being on a similar path, only difference being he was still in the fact-finding stage while she was putting her plot into motion. Since Claude's go-to scheme - aside from mild stomach irritants - is the sudden appearance of a ton of Almyra troops, I don't believe his eventual plot would have been much more peaceful.
The fact that Claude does hold ambitions and builds his power to fulfill those ambitions is why giving them all up in Azure Moon strike me as editorial fiat so Dimitri can have his cake and eat it too. He could have just... not? Nothing was pressuring him to give up anything and Dimitri saving him allowed him to keep everything he had. But he suddenly is feeling supremely magnanimous towards the guy who likely killed a bunch of his friends at Groder and gives him everything as a thank you gift for solving a problem that he caused in the first place. At least in Crimson Flower he is given the dignity of putting up a fight to keep his power.
Regarding gifting Failnaught, while in Crimson Flower it's implied he did surrender it,
only for Arundel to come confiscate it, more importantly it fits into the meta narrative of how relics, or the lack of them, fit in to the theme of the different routes. Relics are the embodiment of the crest system. It's the inherited power of the noble bloodlines made literal and that power is what is used to uphold and justify those noble bloodlines in turn. Fitting then, that Azure Moon is the route where all the Heroes' Relics are available to the player. The tragedy of the Blue Lions is that, intentionally or not, they fight to preserve the system that hurts and dehumanizes them all, and use the tools of that system to do so. Fittingly in contrast, the Black Eagles themselves have no Heroes' Relics, though instead they have access to Agarthan weapons.