Midland63 The L/G is kept down during the wind shear escape routine (also) because terrain impact is a possibility. The loads absorbed by the collapsing undercarriage structure make survival of the occupants more likely, any straw counts.
While the doors protruding into the airflow will add drag, and the explanations above that it is accounted for in the EFATO case are sound, the mainstay B737 does not have any on the main landing gear. Go figure? The priority on both GWPS (terrain) and WSHR (loss of lift) scenarios must be to fly the aeroplane up, as the single objective. I gather that history teaches us messing with configuration might be detrimental to the crew's coordinated effort to ensure max thrust is set and peak climb performance is established (with speed-brakes retracted).
My best guess is that avoiding the doors' extra drag applies for GPWS, where the danger of impact is truly immediate. For WSHR, the impact-survival aspect would be the reason, which covers the 737 too. As well you want to keep the procedures simple and similar as much as practicable. What is seen in the video may be better described as a low-level G/A with unavoidable touchdown due to a de-stabilized flare, owing to gust (most likely). The industry standard wind shear model is probably different, though related. Still, the L/G lever is not to be touched!