Aircraft spares make the supply ship a carrier-group vulnerability

Would a larger carrier with room for more spare parts solve the problems of operating several aircraft types?

It would help with storage, but the different types would still demand extensive spares and nearby logistical support.

A larger carrier would have more room for spare parts, but supporting several aircraft types would still require a large and varied stock. American carrier groups depended on nearby fast supply ships to help sustain those aircraft. Losing that support could leave the air wing short of spares after only a few days of operations. The carrier's ability to keep flying therefore depends partly on a separate ship that an opponent could target.