Your question is too broad, but I think that the work of Grothendieck and Cinsinski (and of course, Maltsiniotis) should be mentioned here. Grothendieck introduced the notion of test category in Pursuin stacks (will we ever be able to fully exploit the great amount of ideas contained there?). Those are the categories that presheaves of sets over them produce models for homotopy types of CW-complexes. Therefore that is the answer to what is special about $\Delta$ (and also an answer to what is not, since there are tons of test categories). Cisinski showed that presheaves of sets over a test category have a model structure. He has an excellent book on this, that you can download from:

http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf

Your question is too broad, but I think that the work of Grothendieck and Cinsinski (and of course, Maltsiniotis) should be mentioned here. Grothendieck introduced the notion of test category in Pursuing stacks (will we ever be able to fully exploit the great amount of ideas contained there?). Those are the categories that presheaves of sets over them produce models for homotopy types of CW-complexes. Therefore that is the answer to what is special about $\Delta$ (and also an answer to what is not, since there are tons of test categories). Cisinski showed that presheaves of sets over a test category have a model structure. He has an excellent book on this, that you can download from:

http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf