We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac1n$, and examine the asymptotic behaviour, as $n \to \infty$, under weak asymptotic assumptions for the (rescaled) particles on row $n$: The empirical distribution of these converges weakly to a probability measure with compact support, and they otherwise satisfy mild regulatory restrictions. We prove that the correlation kernel of particles in the neighbourhood of `typical edge points' convergences to the extended Airy kernel. To do this, we first find an appropriate scaling for the fluctuations of the particles. We give an explicit parameterisation of the asymptotic edge, define an analogous non-asymptotic edge curve (or finite $n$-deterministic equivalent), and choose our scaling such that that the particles fluctuate around this with fluctuations of order $O(n^{-\frac13})$ and $O(n^{-\frac23})$ in the tangent and normal directions respectively. While the final results are quite natural, the technicalities involved in studying such a broad class of models under such weak asymptotic assumptions are unavoidable and extensive.