After discussing this with Tim we came up with the following answer:

The first steifel whiteny class $\omega_1$ of $M$ can be written as the following composition:

$$M \to BO(n) \to BO \to BAut(\mathbb{S}) \to BAut(\mathbb{Z}) \simeq B\mathbb{Z}/2$$

But if $M$ is of type $\ge 2$ then $[M,BO]\simeq [\Sigma^\infty M, bo] \simeq 0$ since $bo$ is of height $\le 1$. So $M$ must be orientable in cotradiction with the third point. 

**Conclusion:** All closed smooth manifolds are of type $\le 1$.



<s>Oh and I believe that at odd primes, type $1$ complexes can be realized by Lens manifolds.</s> Here I was uncareful. This is wrong as it conflicts with the third point of tim as pointed out by Gregory Arone in the comments. So the odd prime type 1 case is still unanswered.