Wind affects your flight path angle by affecting how far you travel while you're descending.
With no thrust from the engine an airplane is falling out of the sky at a roughly constant rate of descent. The wind direction affects the groundspeed, and thus the distance covered between when the engine failed & when the aircraft hits the ground.
This is more directly observable from outside the airplane.
Consider these 4 scenarios on a post-it note: The airplane has lost its engine, and the pilot is flying at best-glide airspeed while holding a constant magnetic heading (making no wind corrections):
The aircraft descends along the no-wind glide path (about 45 degrees in my drawing), and travels in a straight line over the ground (the "ground track") while doing so.
The aircraft descends along a steeper glide path (about 60 degrees in my drawing), because while it's still descending at the same rate the headwind is slowing its progress over the ground. Since it's a direct headwind (right on the nose) the ground track is still straight.
The aircraft descends along a shallower glide path (about 30 degrees in my drawing), because while it's still descending at the same rate it has a tailwind, which is making it move faster over the ground. Since it's a direct tailwind the ground track is still straight here too.
The aircraft descends along the no-wind glide path (because it has neither a headwind nor a tailwind), but the crosswind pushes the ground track (in this case into the inconveniently-placed tree).
Your endurance (time aloft) will be similarly unaffected: If you've set the aircraft up at its optimum glide angle-of-attack (best glide airspeed) the descent rate is pretty much the same regardless of whether you have a headwind, tailwind, or crosswind. You will fall the same distance in the same amount of time, it's just a matter of how far you travel while you're falling (which determines where you land).
You can test this for yourself in an airplane by doing engine-out drills: Line up with a headwind, pull the throttle out, and pitch for the aircraft's best glide speed. Note your vertical speed (rate of descent), and then repeat the test in the opposite direction (tailwind) or at 90 degrees to your original course (direct crosswind): The vertical speed will be effectively the same.