You will experience an apparent headwind when sailing on a broad reach faster than the wind, yes. But the jibe involves turning downwind (thus losing lift), as

You will experience an apparent headwind when sailing on a broad reach faster than the wind, yes. But the jibe involves turning downwind (thus losing lift), as well as bringing the sails across (which will temporarily de-power them). It's been a while since I've been sailing, but I don't think most boats can maintain an apparent headwind through the entire maneuver, even if they are capable of faster-than-wind speed. Some can, which is why I mentioned downwind tacking for high-performance boats, but I think it takes a pretty fast boat to manage it.