Is it possible to create (m)any theoretically workable framework(s) - that do(es) produce probabilities - by taking QM and replacing the Born(-like) rule(s) with something that is not equivalent to it (them)? And, only if so, would that still be called a quantum theory (not necessarily physical such as QM) or does one call that something else?
I would imagine that this is not possible, at least not without making such radical changes to the other postulates that it can no longer be reasonably said to be "a quantum theory". The reason is that if you express quantum mechanics in terms of the density matrix formalism (rather than in terms of pure-state wavefunctions) then the Born rule becomes "the probabilities are the diagonal entries of the density matrix." (Or, more generally, "probabilities are the results of linear operators that project the density matrix onto $[0,1]$.")
The trace of a Hermitian matrix is preserved under unitary transformations, and it is this that leads to the conservation of probability in this formalism. I suppose you could try to change this rule to something else, like "the probabilities are the (normalised) cube roots of the diagonal entries of the density matrix", but the problem with this is that these new "probabilities" will no longer combine like probabilities when you combine density matrices.
So it seems that if you want to keep the ability to formulate quantum theory in terms of density matrices, it probably isn't possible to change the Born rule without messing with either (a) the rules of classical probability theory, or (b) unitary evolution. Changing either of these would be extremely radical, and would probably result in needing to change almost everything else as well.
I object to the people who say if a theory doesn't give different results, "why bother"? It's become fashionable in physics to snicker at the idea of looking for physical meaning behind the equations; this same point is made as the ultimate identifying mark of a crackpot in John Baez's semi-satirical "crackpot index".
If this were really the mark of worthlessness in physics, why would we not snicker at Lagrangian Mechanics? Does it make any different "predictions" than Newtonian Mechanics? And if not, does that make it useless?
You cannot tell if one person is smarter by another by looking at the things those people claim to find interesting. In particular, the statement that you do not find a new theory "interesting" because it fails to make different predictions does not make you smarter than me.
Although I don't really understand it, I think that Bohmian mechanics strives to explain the evolution of systems without invoking the Born postulate. I'm not a fan of Bohmianism because I don't find it physically convincing; I lost interest in it when a Bohmian told me that one possible configuration of the hydrogen atom had the electron sitting stationary a fixed distance from the proton. But that's just me.
I believe that ultimately we have to be able to explain things in physics by the causal time-evolution of some kind of wave function in good old four-dimensional space time. I've succeded in creating causal wave-theoretic explanations of a handful of phenomenon which are traditionally held to be examples of wave-function collapse, including the appearance of flecks of silver on a photographic plate illuminated by the light of a distant star. You can read my explanation on my blog under the title "Quantum Siphoning".