Hi Jacek,
I have been curious about the role of LQG. I will confess that I am primarily oriented towards the string theory perspective. I do think LQG has some relevancy to physics, but it is uncertain what that is. LQG comes from the ADM approach to general relativity, which give constraint equations NH = 0 and N^iH_i = 0 with no explicit time dependency. The lack of time dependency means that energy is not defined. This is a manifestation of Gauss' law, where on a general manifold there is no boundary from which to integrate over to define mass-energy as the source of the field. So the Wheeler DeWitt equation, and spinor variations on that theme in LQG, are in effect constraint systems. The problem is that we do not know what this contrains exactly.
I have thought that LQG is some sort of "target" of a renormalization group flow in string/M-theory. However, string theory has at its IR limit a graviton in a weak coupling regime on a background. This theory is renormalizable as a perturbative field theory, even if we know it is a weak coupling approximation to quantum gravity. LQG is not renormalizable. So we are sort of left with an open question. Is LQG a strong coupling S-duality to the string weak coupling theory? There are some problems with an idea of this sort. In particular LQG is not easily embedded into a larger unification scheme with gauge fields or supersymmetry.
However, LQG is based upon basic general relativity in a way that is hard to ignore. I suspect it is not completely wrong, as is often thought in the string theory community. However, where it fits into things is unknown.
LQG is related to Regge calculus, which is run on a computer. It is one of the tools numerical general relativists use. However, this approach to general relativity is also a manifestation of the action principle. Underlying it all is an extremal principle. Extremization is a continuous process and from a set theory perspective it is a manifestation of a nondenumerable set of numbers, or the reals. This is one motivation to suggest that the universe is not entirely a computer, even if it has some computer or algorithmic-like structures to it.
Cheers LC