You are not missing something. Multiclassing into fighter only gives light and medium armor proficiency (as well as shields and simple and martial weapons). You would have to also take the Heavily Armored feat to get heavy armor proficiency.
Somewhat oddly, while multiclassing into cleric only gives light and medium armor plus shields, it also gives you full access to the cleric domains, several of which give heavy armor proficiency. It doesn't seem quite fair that fighters don't get that, but there it is.
Perhaps fighter was considered too attractive of a "dip" with fighting style and second wind also at first level and very attractive additional features at second and third. But that's just speculation. Whatever the reason, if you want heavy armor proficiency from fighter (or paladin), you need to take that as your first-level class.