  We consider the simple random walk (or P\'olya walk) on the one-dimensional
lattice subject to stochastic resetting to the origin with probability $r$ at
each time step. The focus is on the joint statistics of the numbers
${\mathcal{N}}_t^{\times}$ of spontaneous returns of the walker to the origin
and ${\mathcal{N}}_t^{\bullet}$ of resetting events up to some observation time
$t$. These numbers are extensive in time in a strong sense: all their joint
cumulants grow linearly in $t$, with explicitly computable amplitudes, and
their fluctuations are described by a smooth bivariate large deviation
function. A non-trivial crossover phenomenon takes place in the regime of weak
resetting and late times. Remarkably, the time intervals between spontaneous
returns to the origin of the reset random walk form a renewal process described
in terms of a single `dressed' probability distribution. These time intervals
are probabilistic copies of the first one, the `dressed' first-passage time.
The present work follows a broader study, covered in a companion paper, on
general nested renewal processes.
