An std::array<> has no constructor that takes an std::initializer_list<> (initializer list constructor) and there is no special language support for what it may mean to pass a std::initializer_list<> to a class’ constructors such that that may work. So that fails.
For it to work, your derived class needs to catch all elements and then forward them, a constructor template:
template<typename ...E>
enum_addressable_array(E&&...e) : base_t{{std::forward<E>(e)...}} {}
Note that you need {{...}} in this case because brace elision (omitting braces like in your case) does not work at that place. It’s only allowed in declarations of the form T t = { ... }. Because an std::array<> consists of a struct embedding a raw array, that will need two level of braces. Unfortunately, I believe that the exact aggregate structure of std::array<> is unspecified, so you will need to hope that it works on most implementations.