Regarding your "simple observation", I think you meant to say: [...] all circles of radius $d/2$ [...] square of length $1+d$ [...].

One question of clarification: You explain how the sphere-packing problem and the maxmin problem are related but section 2 of this reference states that they are equivalent(?). Can I get the maxmin solution from a densest packing of spheres?

Since I was interested more in the general case (k not necessarily large), I found your Locatelli & Szabó references particularly useful. I'll wait a little more before accepting this as the answer.

I'm interested in the continuous hypercube, thanks for asking.