Swords are not that good on Edelgard because Aymr is not a sword. You cannot be serious here, dude.
Edelgard has enough HP to not get one-rounded in the early game (especially with HP+5 which you can get by Chapter 3) on the off-chance you miss a 95% Smash in 2RN.
You pick up an Accuracy Ring in chapter 5 from a Knight if I recall correctly. They might be "rare" but if Edelgard is ending chapters in two turns reliably with it I would say it is a more worthwhile investment than putting it on someone like Hubert so he can hit Mire more and still not use Raging Storm. Same deal with Empire Pegasus battalion - yeah, another flier COULD use it but they aren't Edelgard, so they won't use it as well. Ending maps in 2 turns offsets any opportunity cost for a unit not named Edelgard in CF. Authority ranks are shockingly easy to build so I don't really understand why that is a point of contention.
Axe Prowess 3 or 4 gives her enough hit to reach reliable numbers against the enemies that matter, none of whom are Assassins or Swordmasters since you really only care about hitting bosses, not to mention the hit boost from linked attacks if seeing a 9 as the first number of your hit rate scares you so much.
In a slower playstyle, sure, it isn't as broken because you are consciously trying not to end chapters in two turns with it, but to say that the strongest strategy in the route is "unreliable" and therefore not that good is just being a contrarian for the sake of it, Levant. Go try it yourself and see just how stupidly good it is.