You know, making a character whose entire shtick is being a "Master Tactician" blood-lusted kind of defeats the purpose...
Anyway, Edelgard matches many of Claude's hax. Her Seal Attack and Seal Magic match his Seal Defense, while his Silence and Hexlock Shield all no-sell Edelgard's canonically more potent magic. Edel's Grounder's effective damage is negated by Aurora Shield. Edelgard's Sword of Seiros matches Claude's Sword of Belgata (and outright surpasses it if she deals damage to him in the exchange), and Pavise no-sells Claude's close-range sword attacks if he's suicidal enough to get in close. They're also pretty even in healing options.
Again, just like with Nemesis, if Claude rushes Edelgard in close-combat, he gets rekt since that's her bread-and-butter and she likely breaks him like a Kit-Kat bar. Both Claude and Edelgard are bloodlusted, but that doesn't make them idiotic. It really just means that, if they can abuse their winning move, they will do so without suffering from PIS (otherwise Claude gg's by telling Edelgard there's a rat in her armor). And Claude's best moves are not close-ranged, but long-ranged.
Edelgard has Counterattack, but it makes it so she can only attack after Claude does, so he will always have the battle-momentum if she tries to abuse that. She has her Tomahawk, sure, but throwing a Tomahawk basically removes her chances of using Aymr's Raging Storm for her to try to migitate their distance (and Claude's Curved Shot, Fallen Star, and Wind God also outrage that). And if she really started to get close, he can go for Encloser to immobilize her momentarily.
Meanwhile, Claude has Failnaught's Fallen Star, which he will likely abuse from the get-go since, honestly, its dodge hax is pretty broken. Even if it runs out of uses, he also has other high-damage skills. Using Fallen Star, Wind God, and Curved Shot ensure Edelgard never gets to use her best, and close-ranged, moves all while doing considerable damage to Edel. His wyvern makes him have MUCH better mobility, speed, and accuracy (Edel does have accuracy hax skills to mitigate her low accuracy axe skills but even then Claude is much faster and has skills to dodge as well). Silence and Hexlock Shield cripple Edel's attempts to capitalize on Claude's paper-thin Resistance with her magical girl power.
EDIT: If we consider that Claude cannot use magic for whatever reason, then if he gets hit by a Resistance-negating Luna (which can bypass Hexlock Shield) or a powerful Hades attack (which does not, but it's pretty powerful nonetheless), then he's likely getting nuked. It's a valid win-con for Edel. And yet, it only has 1-2 range while some of Claude's best attacks can go up to 5. Gameplay mechanics aside, it's a clear indication that he can outrange her best magical attacks and she'd have to rely on getting close/work with Counterattack. And that's not considering Fallen Star's dodge hax, which would be troublesome for Edel to deal with.
Simply put, Claude's options to take down Edelgard are simply better by virtue of his aces all being long-ranged whereas Edelgard's best options are either close-ranged (since she is SilencedÔäó) or not ranged enough. It is just hard to believe using Counterattack the entire fight will do anything but delay the inevitable. She does have win-cons, but in most instances Claude's win-cons are simply more reliable.
tl;dr voting Claude