A basis is required to be both spanning and independent. Why does dropping either requirement break the thing a basis is for - giving every vector exactly one set of coordinates?

D. Because spanning gives coordinates and independence makes them unique.

The two conditions do different jobs and each failure has its own symptom. Drop spanning and (0,1) cannot be written in terms of (1,0) at all. Drop independence and (2,1) has infinitely many representations over {(1,0), (0,1), (1,1)}. Together they give existence and uniqueness, which is what lets an abstract n-dimensional space be treated as a list of n numbers.