Well it’s not “bad” it just makes travel slower in cruise.
A headwind does push more air over the wings like you said, which creates more lift. This is good for takeoffs, as the plane can takeoff at a lower speed and use less of the runway, as well as climb to altitude faster. It’s also good for landing for the same reasons.
But in cruise, the extra lift doesn’t really do anything. Going against the wind makes the plane’s ground speed slower, so the flight will take a bit longer. Where as a tail wind in cruise is effectively helping push the plane forward, so the flight will be shorter.
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