Why does the $rm H$-burning Shell do the $rm CNO$ cycle instead of the $pp$ Chain?

At the end of core hydrogen burning in a low-mass star you have an inert He core that is essentially isothermal and hotter than the temperatures required for H-burning. H-burning starts at the edge of the core and this is hot enough for the CNO cycle to dominate.

Why does core contraction halt? No energy is generated in the core so it will keep contracting and getting hotter (by the virial theorem) until the energy losses outward are balanced by inward flux from the shell burning. But this doesn't happen until the burning rate is high (higher than on the main sequence - the luminosity of subgiants and giants is higher than for main sequence stars) and occurs in a relatively thin shell (because of the temperature gradient). Hence the temperature of this shell needs to be higher than the core of a main sequence star of the same mass and so the H-burning occurs via the CNO cycle.