Think of the radial nerve as splitting at the elbow into two streams: a sensory stream (superficial radial) and a deep motor stream (posterior interosseous nerve). A low radial palsy hits only the deep motor stream after the split. To answer, ask of each option whether its supply leaves the trunk before or after that split.
The extensor carpi radialis longus receives its twigs from the main radial trunk while it is still in the arm, well above the elbow split, so a low lesion cannot reach it and it keeps working. The extensor carpi radialis brevis and all the long finger extensors are downstream targets of the posterior interosseous nerve, so they fail and the patient shows finger drop. Cutaneous supply to the back of the hand travels with the superficial radial branch, which also peels off above the lesion, so this sensation is intact; the posterior interosseous nerve carries almost no skin sensation anyway. The only listed structure whose nerve supply arises proximal to the injury and is therefore spared is the extensor carpi radialis longus.
\[\boxed{\text{Extensor carpi radialis longus}}\]