UPDATE: It's been pointed out to me that older work by Jim Carrell and Dale Peterson involves the non-negativity condition, though their main goal is the study of singularities of Schubert varieties in classical cases. See the short account (with a long title) by J.B. Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties.Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), 53–61, Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc., Providence, RI, 1994. The first section develops for an arbitrary Coxeter group some consequences of non-negativity of Kazhdan-Lusztig coefficients for the combinatorial study of Bruhat intervals. (For further details about the geometry, see the 2003 paper by Carrell and Kuttler in Invent. Math. 151.)
I'm still not sure whether such consequences of the 1979 K-L conjecture are enough to make the conjecture in itself "important". But it's definitely been challenging to approach.