We establish the existence of complete K\"ahler metrics of semi-positive holomorphic sectional curvature with many zeroes in an interesting and natural geometric setting. Specifically, we use Calabi's Ansatz in the form due to Koiso-Sakane to produce such metrics on the total space of powers of the tautological line bundles over the projective line. We formulate a general conjecture regarding the compact case and use a product approach to obtain an analogous result in the non-compact case.