I don’t agree.
The HK1 and HK2 are definitely related (second one “built” upon the first), yet they can stand as separate theorems by themselves.
The first theorem states that there is a unique density for every external potential, but it does not imply the existence of a variational principle over densities. The second theorem introduces the minimisation of energy functional to find the ground state density, which is a different and an independent principle, yet one that is an essential addition.