I think mages should be capable of both, but not necessarily at the same time. This I think is where 3.5 and PF messed up, it's too easy to make a mage that is all things to all parties with little investment other than picking the right spells to put in their many slots.

You should be able to make the archer mage (though I personally prefer the term "artillery mage") if you like, and you should be able to make an Awesome By Analysis controller if you like too. You should also be able to make the Gandalf "swordmage" (or "clawmage") if you want, or the Pokemon Master mage. What you should not be able to do, is switch between them easily during the day, or even by merely taking a night's rest. Building for one should make doing the others more difficult.

But mages that are only DPS-bot archers are not fun and belie the cunning that the archetype is meant to represent. Anyone who has watched Log Horizon knows what I'm talking about - the protagonist is a God Wizard who is very weak on his own, but gets his entire party to perform as a well-oiled killing machine through his mastery of the system.