Yes you could still *do some* epic quests without an arcane (though the ones I've done pre-U9 without an arcane had a divine using greater command, and divines are casters...), the usage of resources in such a run will often be many times higher than in a run that does have an acane.
But the fact is, that outside the very best players, arcanes basically were essential. And even then, there were, and still are, quests where they practically are essential even for the elite players. Sure you can run quests without arcanes or divine evokers (no, healbots don't count), but there are many quests where you'll be spending over twice as much time as you would if you have them with you. And if you take a few more good casters in a difficult quests it'll be much, much smoother than if you had taken a couple good melees instead.
So no, even if it is possible to run quests without arcanes, the resource consumption and extra time spent just reinforces my point: Arcanes were, and are a requirement for a smooth epic run. You can run Phiarlan and Deneith epics even without arcanes, but it won't be nearly as smooth or fast if you had taken the arcane, and spent resources are going to skyrocket. Good luck trying deserts without arcanes or evokers (or crazy "bards"

).