The tin cans of that era are a far cry from their modern descendants. They had actual armor, where a modern DD generally doesn't.
Also, most of the weight of a shell is the shell, not the warhead/payload. The Silkworm actually carries more explosive than a 14" shell.
It wasn't a Silkworm, but I still think that you're right. The missile was carring 363 lbs of explosive. I had to look it up:
(NAVAL ORDNANCE 1937 CHAPTER XIII PROJECTILES)
"Where the projectile must meet heavy face-hardened armor, the result is a massive piece of steel with a heavy head, thick walls and a small cavity, called an armor piercing projectile. In such projectiles the weight of the bursting charge (high explosive) varies between 2.1 per cent to 2.6 per cent of the total weight of the projectile. If the size of the charge is increased the Projectile is unequal to matching even caliber plate at battle ranges and angles of fall, while if the size of the charge is reduced the fragmentation of the projectile on burst is not efficient. (Plate I, Fig. 3.)
Where the projectile may encounter armor of a thickness of only to 1/2 caliber, or a combination of light side plating, bulkheads, and decks, the cavity can be somewhat enlarged, and the design is called a common projectile. In such projectiles the charge represents, for major and intermediate calibers, about 6.0 per cent of the total weight, and for sizes below 6 inches the percentage falls to about 3.0 per cent to 3.5 per cent. (Plate I, Fig. 4.)"