Edge isn’t tsundere. Not really. Tsundere will pretend not to like someone or simply aren’t equipped to deal with their own feelings and will lash out rather than be honest with themselves and others.
No, Edge isn’t tsundere. He’s just really bad at flirting. He hasn’t had much, if any, practice himself, and what experience he does have is…less than positive. So unless someone is very direct about their interests, it tends to go over his head. Given his gruff nature, it often comes off as if he isn’t interested or even as a harsh brush off. But, honestly, he just doesn’t necessarily know that someone is trying to flirt with him.
Imagine:
“Hey, I’m seen you around town, and I was thinking–”
“Really? Because I’ve never noticed you.”
Or:
“That fall from heaven had to hurt, angel.”
“…I am a monster, not a devil. Please educate yourself, human. There is a difference.”
Edge isn’t tsundere. He just doesn’t know how to flirt.