Why an Alaska-class ship could not remain original today

Could an Alaska-class large cruiser still be in active service today?

Only after such extensive rebuilding that it would effectively be a new ship with the old name.

A large cruiser can be modernized repeatedly, and a surviving Alaska might even receive nuclear propulsion. The original hull and keel still accumulate fatigue, corrosion, and structural wear over decades. Keeping the ship in commission to the present would eventually require replacing so much structure that the material continuity becomes nominal. A modern Alaska would therefore be a new vessel carrying the old nameplate rather than the original ship preserved unchanged.