Why similar frigate designs can emerge at different times

Does the convergence of frigates and destroyers on the Royal Navy's approach mean Britain alone had the right idea?

No; other navies could reach the same design logic, while different building cycles determined when they implemented it.

Several navies can see the same logical direction in frigate and destroyer design. Their construction cycles determine when they have the opportunity to turn that idea into ships. The Royal Navy's earlier adoption is explained in those terms, without implying that other navies lacked the same idea. Similar later designs can therefore reflect shared requirements and different programme timing.