But the question here is not looking for a generator, because it would be okay if some group elements are only reached during the application of the sequence. (For the sequence to be a generator, all group elements need to be reached at the end of some full application of the sequence.)

The Hamiltonian cycle sequence from the original post is not a generator, but it visits every state. The question is: Is there a significantly shorter sequence that (when repeated) does the same?