I can't imagine any kind of curvature condition which would work. I will use the phrase "standard examples" to refer to a space diffeomorphic to a sphere or projective space over $\mathbb{C},\mathbb{H}$ or $\mathbb{O}$. This are certainly the most well known examples of manifolds having the kinds of cohomology rings you are trying to avoid.
For example, one could demand negative/zero/positive sectional curvature. But the first two cases can't arise on a closed simply connected manifold, by Cartan-Hadamard Theorem. This leaves the case of positive sectional curvature - but then all of the standard examples admit such a metric.
So, perhaps one could demand negative/zero/positive Ricci curvature. Again, in the positive case, all the standard examples admit such a metric. Further, Lohkamp has shown that every manifold of dimension at least $3$ admits a metric of negative Ricci curvature. And it seems to be open whether or not $S^n$ admits a Ricci flat metric (see this MO question). I do not know about Ricci flat metrics on any other space with singly generated rational cohomology.
Finally, in the realm of scalar curvature, all the standard examples admit metrics on positive scalar curvature. By Kazdan-Warner, they each admit scalar flat and scalar negative metrics.
To further complicate matters, there are known examples of exotic spheres which do not admit metrics of positive scalar curvature (so don't admit metrics of positive sectional or Ricci curvature). So it doesn't seem like any of the usual curvature conditions can limit the rational cohomology ring like you would like.