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You’re mixing two separate design elements. Hull design and rig design.
Schooner, sloop, ketch, yawl and lateen discribe mast and sail configurations. Hull design is separate from the rig. What you are discribing is a double ended schooner. The hull could be a narrow canoe or wider bigger boat like a Tancook Whaler.

Rigs and sail plans are matched to hull designs for various reasons. Schooners were used to allow many smaller sized sails and standing rigging that are more easily controlled than what the same boat would have using a sloop rig. Ketch and yawl rigs are easy to handle an the mizzen (the shorter mast aft) keep the boat from falling off the wind. They point up well.