  (Abridged) We use N-body simulations to study the effects that a divergent
(i.e. "cuspy") dark matter (DM) profile introduces on the tidal evolution of
dwarf spheroidal galaxies (dSphs). Our models assume cosmologically-motivated
initial conditions where dSphs are DM-dominated systems on eccentric orbits
about a host galaxy composed of a dark halo and a baryonic disc. We find that
the resilience of dSphs to tidal stripping is extremely sensitive to the halo
cuspiness; whereas dwarfs with a cored profile can be easily destroyed by the
host disc, those with cusps always retain a bound remnant. For a given halo
profile the evolution of the structural parameters as driven by tides is
controlled solely by the total amount of mass lost. This information is used to
construct a semi-analytic code that simulates the hierarchical build-up of
spiral galaxies assuming different halo profiles and disc masses. We find that
tidal encounters with discs tend to decrease the average mass of satellites at
all galactocentric radii. Interestingly, satellites accreted before
re-ionization (z>6), which may be singled out by anomalous metallicity
patterns, survive only if haloes are cuspy. We show that the size-mass relation
established from Milky Way (MW) dwarfs strongly supports the presence of cusps
in the majority of these systems, as cored models systematically underestimate
the masses of the known Ultra-Faint dSphs. Our models also indicate that a
massive M31 disc may explain why many of its dSphs fall below the size-mass
relationship derived from MW dSphs. We use our models to constrain the mass
threshold below which star formation is suppressed in DM haloes, finding that
luminous satellites must be accreted with masses above 10^8--10^9 M_sol in
order to explain the size-mass relation observed in MW dwarfs.
