Warship stern forms follow speed, stability, and load requirements

Why did British capital ships use rounded sterns while many cruisers, destroyers, and torpedo boats had sharper sterns?

The stern was shaped for the ship's required balance of load, stability, speed, maneuverability, and efficiency.

No single stern form is inherently more modern or seaworthy than another. A large armored ship may favor a rounded stern to support weight and stability. Faster cruisers, destroyers, and torpedo craft can use sharper forms suited to speed and maneuverability. The required mission and the shape of the entire hull determine which compromise is appropriate.