I think there's a fallacy in the bike story problem. A bike in a crosswind is not like a boat in a cross current because the bike drives on solid ground and can be steered to maintain a straight path. If the bike is not moving in the crosswind direction, how much work is the crosswind doing on the bike?

I would agree that it's true that most crosswinds are still bad winds that slow you down, but I think it's due to more complicated factors than the simple vector addition in the physics professor's example.