Churchill's opposition to Overlord as coalition bargaining

Did Britain win in spite of Churchill's Mediterranean focus and opposition to Overlord and Dragoon?

Some of Churchill's opposition can be read as bargaining: resisting an operation that would proceed anyway could secure more for Britain than immediate agreement.

Open support would let the United States define Overlord's arrangements without paying a political price for British cooperation. By resisting and emphasizing Mediterranean alternatives, Churchill could seek resources or concessions before accepting a cross-Channel operation he also wanted. This reading presents coalition strategy as negotiation among allies as well as planning against the enemy.