I somehow disagree. Life isn't always as beautiful as the homotopy type of a CW complex especially if one is not interested in algebraic topological questions for its own sake.

Thank you for your second editing, which was enlightening for me.

The corrections to the McDuff-Segal paper Peter May mentioned can be found as Lemma 3.1 in D.McDuff - "The homology of some groups of diffeomorphisms.".

arxiv.org/abs/math/0610216

Minor correction: "[...]whose $\alpha$-invariants (as spin manifolds) are NONzero."

Does one not even need finite dimensional cohomology and not just finitely generated? The result is usually proved by the classification of finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0. A theorem of Hopf says that it is a free exterior algebra with generators of odd degree. Does it really suffice to have finitely generated cohomology?

As a supplement: More general is every $(k-1)$-connected compact manifold of dimension $\leq (4k-1)$ formal for $k>1$. The result mentioned by Sinan Yalin follows for $k=2$. To get a counterexample in dimension 7, one can (as an alternative to the mentioned papers) realize (via Sullivan-Barge-Realization) the minimal Sullivan algebra $(\bigwedge<v,w,x,y,z>,d)$ where $deg(v)=deg(w)=2$, $deg(x)=deg(y)=deg(z)=3$ and $d(v)=d(w)=0$, $d(x)=v^2$, $d(y)=vw$, $deg(z)=w^2$ as a compact simply connected 7-manifold. This result will be non-formal, since $<v,v,w>$ is a nontrivial Massey product.