Why the Admiral Hipper class remained heavy cruisers

Were the Admiral Hipper-class ships essentially heavily armed anti-aircraft cruisers despite their eight-inch guns?

No. They were heavily overengineered eight-inch heavy cruisers.

A conspicuous anti-aircraft battery did not redefine the whole ship. The Admiral Hipper class remained a group of eight-inch-gun heavy cruisers. Their design is characterized here as heavily overengineered as well as heavily armed. Warship classification should follow the main armament and overall type instead of one secondary weapons fit.