The confusion arises because there are essentially two different approaches to the LQG dynamics in the literature.
The canonical approach pioneered by Thiemann is to define the matrix elements of the Hamiltonian constraint operator acting on spin networks, and then postulate the kernel of that operator to be the physical Hilbert space of LQG. It is extremely hard to make any physical predictions with this mathematical object as it is so extremely complex. However, it is possible to prove that it is mathematically well defined.
The spinfoam approach is based on the so-called EPRL spinfoam model, which is what the paper you linked talks about. It indeed reproduces General Relativity (or rather its triangulated version – the Regge theory, which becomes General Relativity when the triangulation is refined) in the $\hbar \rightarrow 0$ limit if the boundary states are taken to be the coherent (semiclassical) Livine-Speziale states. Rovelli has a nice chapter with a detailed calculation in his textbook called "Covariant Loop Quantum Gravity".
The spinfoam formulation, unlike the canonical formulation, has not been made into a mathematically well-defined and consistent formalism so far. Due to infrared divergences, it isn't clear whether the projective limit of spinfoam amplitudes exists or not, and whether, if it does exist, it specifies a projection operator onto the physical state space of quantum gravity.
This situation is a lot like the one in perturbative QFT. The theory gives meaningful physical predictions, but so far no one has been able to prove that this theory exists mathematically.