If opening up Doran's Ring to other users is the goal

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The whole point of 3 starter items is: Shield for the manaless or tanks, Ring for the casters and mages and Sword for AD reliant champions. Champions that rely on attack damage shouldn't need the mana and vice versa, specifically in the early game. Making it adaptive just defeats the point of having multiple items in the first place.

No it doesn't? If you're worried about starter items existing for specific classes only, why not take it away from a class situation, and move it to a playstyle situation. Decide how you want to play in lane, not be forced to take this starter item because "I'm a mage and this is the mage item"