Essentially the same question has been answered; see http://mathoverflow.net/questions/242448/generalizations-of-the-handle-trading-techniques/242466#242466.  For simply-connected manifolds, Smale showed that you can get a handle decomposition that is as simple as required by the homology groups. (The spin condition is not relevant at all.) A handle decomposition gives a homotopy equivalent cell complex; you won't get any simpler than this.