Why navies seek sea control in the first place

What missions do Western navies perform besides sea control, including work such as icebreaking?

Those missions make sea control useful by protecting commerce, opening routes, and enabling supplies and forces to move.

Sea control matters because of what a navy can do with it. Control allows trade, supplies, and military forces to move across maritime routes, while missions such as icebreaking and fisheries protection keep those routes useful. The China Fleet at Pulo Aura illustrates the economic stakes because its safe arrival could affect the East India Company's fortunes. The Second World War convoy system makes the same point at wartime scale: German submarines inflicted losses, but they never stopped the essential flow of supplies.