Like I could maybe, just barely maybe understand if there was an enchanting perk that made the soul stay there for infinity. But by wearing a piece of clothing that affects your destruction magic? Not buying it what so ever.
Activating a magic item is kinda like using steam to spin a turbine - A novice wastes a lot of that steam in a large, unguided discharge that causes the fanblade to spin slowly because it's undirected. An adept can control the stream, greatly reducing the amount of steam needed to send that turbine spinning rapidly. Someone with 100% Reduction in Magicka Cost is essentially building a pipe around the turbine, not letting any of the steam escape.
For a mage, the source of the steam is their magic pool. For an enchanted item, the source of the steam is the soul. A skilled enchanter can make a more efficient reservoir to hold that mystical steam-anologue, but it still comes down to the user to guide and direct the steam to spin the turbine. Or, to end the metaphor, the end-user uses his own magical skill (Not reserves) to draw upon the magical reserve held within the item (The soul) and engage the power of the item. Enchanted items are no longer "Idiot-proof" like they were in past games.
TL;DR - You're erroneous in your assumption that Magicka-cost-reducing enchantments affect the user's magicka - They actually affect the user's efficiency in directing the energies of that school.