In Schmidt's book Subgroup lattices of groups, Theorem 1.2.5 states that a group $G$ is cyclic if and only if its subgroup lattice $L(G)$ is distributive and satisfies the maximal condition. Its proof uses Ore's theorem stating that the subgroup lattice of a group $G$ is distributive if and only if $G$ is locally cyclic. The proof of this last requires the Fundamental Theorem of Finitely Generated Abelian Groups (FTFGAG).
Question: Is there a proof of Theorem 1.2.5 which does not use FTFGAG?
Remark: In the finite case, such a proof exists (see here with $H=1$).
Motivation: Carmela Musella and Maria De Falco asked me for a relative version of Theorem 1.2.5.