To your last point, is that true though? TWSitD explicitly wield a ton of political influence in the Empire, having the conspirators of the Insurrection of the Seven in their pockets. Edelgard's coup to centralize power would have been impossible without their help, if she stood directly against them she would at best become a puppet as her father did. And would Rhea have ever voluntarily surrendered power to Edelgard, a revolutionary who stands directly against the order that Rhea maintains? In Crimson Flower, Edelgard attacks the Church, and the Kingdom comes to their defence with Dimitri swearing his fealty to Rhea. How could she have avoided involving the other nations in the war?

If there's a better way that doesn't involve compromising her ideals, then I've yet to hear a satisfying explanation as to what it is.