Uniform hierarchy of problems that span complexity and computational hierarchies
[Building off of Kaveh's insight in the comments.] It seems unlikely that someone could come up with a family of problems that is significantly different from quantified boolean formula, without disproving the PH-analog of the Berman-Hartmanis isomorphism conjecture.  Without that, any problem you come up with would be not only equivalent to $QBF_k$, but in fact isomorphic to it. One way to define isomorphism between two languages here is to take a single abstract language, but encode its objects (in this case, quantified boolean formulae) using two different boolean encodings.

On the other hand, isomorphism isn't necessarily a good judge of what's useful for people to come up with proofs.  After all, in the arithmetic hierarchy, Myhill's Isomorphism Theorem proves the arithmetic analog of the BH isomorphism conjecture (in fact, that's history backwards since BH was motivated by Myhill).  Yet, as the question points out, there are several "different-looking" characterizations of various levels, some of which are more useful for proofs than others.

Although it seems unlikely that anyone will come up with such a uniform family of languages for every level of PH, the two surveys (one, two) by Schaefer and Umans discuss natural problems that at least "look different" from QBF for the first few levels of PH.