Direct Numerical Simulations (DNS) of premixed hydrogen-air flames anchored in a fully-developed turbulent channel flow (TCF) are performed at a friction Reynolds number of ${Re}_{\tau }=530$ and thermochemical conditions susceptible to the emergence of intrinsic thermo-diffusive (TD) phenomena acting on the turbulent flame. Two premixed flames are studied: a slower flame ($\phi =0.25$), predominantly propagating within the core flow, and a faster one ($\phi =0.35$), reaching closer to the channel walls and intermittently quenching on it. The present DNS database provides new insights into the characteristics of premixed flames susceptible to TD phenomena and propagating in realistic near-wall shear turbulence. The influence of varying turbulence intensity, and of wall-distance dependent time and length scales, on the flame propagation characteristics is evaluated through a detailed analysis of the local stretch factor $I_0$, quantifying reactivity enhancements caused by TD phenomena. At $\phi =0.25$, the flame response to the fluid motions is mainly forced by the weaker turbulence present in the core flow. This results in an augmented $I_0$ compared to the laminar reference value, suggesting reactivity enhancement by the strongly non-linear interaction of TD phenomena with (relatively) weak turbulent motions present within the core flow. At $\phi =0.35$, as the flame propagates from the core flow towards the channel walls, the flame response is forced by turbulence of increasing intensity, resulting in a corresponding augmentation of the Karlovitz number. Crucially, as the flame propagates into the near-wall region, the peak value of $I_0$ is co-located with the peak Reynolds stresses ($y^{+}\sim 10$). This observation suggests a strong (local) synergistic interaction between TD phenomena and wall turbulence, ultimately resulting in significantly enhanced flame speed.