They absolutely change how the class plays and what tactics the player employs. They do not represent a tweak, they are really must-have for that build the players is going for.

For an easy and cheap example - Smoke and mirrors. You now always succeed and vanish/hide. More expensive assassin ones would be the buki edges, most of them are char-defining.

For other examples: Necro sleep-now-does-damage absolutely changes whether I want to try to sleep that rager now, warrior edges in any spec make certain skills worthwile instead of worthless, AP edges totally change how a char plays post-edge, etc.

On the other hand, there are way too many edges that should be rolled into a class or you just get one per each 5/10 levels to pick for free, like Lucky, Extra X or other niche stuff (You regen better in fire/forest??? Seriously?) that should be the "customize my build" kinda stuff that doesn't cost EP. That's regardless of what you do with the system as a whole.