Why a 250,000-ton carrier adds little AEW value

Would a 250,000-ton carrier gain a meaningful advantage by operating full-sized AWACS aircraft?

No; Hawkeye provides the essential capability without the huge ship and infrastructure penalty.

Hawkeye already provides the airborne warning capability needed for a carrier force. A 767-sized AWACS operates as a node within a different, land-based air-defense system, so its extra size would not give the carrier a proportionate advantage. A 250,000-ton ship would also create major construction and operating difficulties. With current technology, the estimated optimum for a full-sized carrier is about 100,000-110,000 tons.