It depends on what you consider to be an essential, defining feature of the theory. Your question about “explanation” points to the biggest problem: the theory attempted to encapsulate an explanation for all of the facts of phonetics and phonology in the theory of representations. This was a tragic mistake, which we are still trying to eradicate. The slogan of the day – get the representations right and the rules will follow – is simply not true. There has to be a theory of computations, as well as a theory of representations. The general problem with the theory of grammar assumed in ASP is that it became increasingly dependent on a massive list of complex conditions as part of UG. The OCP debacle is an exemplar of the problem, that the OCP can be invoked as an “explanation” for pretty much everything, and as a presumed part of UG is always available “for free”.
However, I don’t consider this to be a problem with ASP as a computational and representational theory, this is a methodological problem that was encouraged by certain ways of thinking that were encouraged in the autosegmental era. The plethora of representational devices (so many competing theories of representation) are not an indictment of the essential claim(s) of autosegmental theory, they are the result of bad methodological practices – which originated in pre-autosegmental practice. The technical weakness that I see is a lack of attention to the theory of computation necessary given the theory of representations, that is, we really do not have a fully fleshed out theory of rules in ASP (or at least, do not yet have one). That’s a real weakness of the theory.
However... the competitors aren't actually in a better position, so it's not a comparative weakness.
