I added my own comment about the situation on the discussion forum of New Scientist and add it also below.
I would decompose the question "Can string theory explain the Universe, or is the concept of multiple universes, each with its own constants of nature and laws of physics, flawed?" to two independent questions since the failure of the string theory need not kill the multiverse concept.In any case, Good Christmas and Better New Year for Everyone,
Some words about my background. I have devoted the last 27 years to develop a generalization of string theory obtained by replacing strings with 3-D surfaces, whose "orbits" in certain 8-D space define space-times as 4-D surfaces (see my home page). I initiated the program 6 years before the first super string revolution at 1984: the motivation came from the conceptual difficulties of general relativity due to the poorly defined notion of energy. This theory has a lot of common with string theories: in particular, the crucial super conformal symmetry. Around 1984 I was certain that it would take a couple of years before string people would realize that super string models do not work as such and ask how to replace strings with 3-D objects. I couldn't imagine that it would take 21 years to see what was absolutely obvious from my personal perspective.
Returning to the questions stated in the beginning. If last 21 years are not enough to convince anyone that string theory cannot explain Universe, nothing is enough.
I however believe that the notion of multiverse makes sense in the many senses that can be assigned to it (quantum universes as quantum superpositions of classical worlds, cosmology as a Russian doll cosmology containing cosmologies within cosmologies, space-time surfaces as many-sheeted structures consisting of geometrically parallel space-time sheets, etc...).
What distinguishes this vision from that of Susskind is that various space-times share common characteristics possessed also by the universe we live in. The dimension of the space-time is four, the symmetry group of fundamental interactions is that of the standard model. Various universes correspond to the spectrum of coupling constants predicted by coupling constant evolution, and the zero modes distinguishing between different universes correspond classical degrees of freedom postulated in ad hoc manner in quantum measurement theory. The view about particle massivation is however different and no space-time super-symmetry is predicted. The theory also predicts a lot of new things, and forces a generalization of quantum theory so that Planck constant is dynamical and quantized: this allows to identify dark matter as a macroscopic quantum phases with large value of Planck constant.
There are good reasons to believe that these predictions, including space-time dimension and the symmetries of fundamental interactions, follow from the mere mathematical existence of the "world of classical worlds" as an infinite-dimensional Riemann geometry required by the generalization of Einstein's geometrization of physics program.
The irony is that the uniqueness of the geometry of the world of classical worlds was discovered in the case of loop spaces (classical worlds are now closed strings) very early by Dan Freed. This marvellous discovery was not however taken as the starting point in attempts to find a generalization of string models by replacing 2-D space-times with 4-D space-times. The probable reason is that the fierce competition did not leave for string theorists time to think about fundamentals.