  We present estimates for the number of supermassive black holes (SMBHs) for
which the next-generation Event Horizon Telescope (ngEHT) can identify the
black hole ``shadow,'' along with estimates for how many black hole masses and
spins the ngEHT can expect to constrain using measurements of horizon-resolved
emission structure. Building on prior theoretical studies of SMBH accretion
flows and analyses carried out by the Event Horizon Telescope (EHT)
collaboration, we construct a simple geometric model for the polarized emission
structure around a black hole, and we associate parameters of this model with
the three physical quantities of interest. We generate a large number of
realistic synthetic ngEHT datasets across different assumed source sizes and
flux densities, and we estimate the precision with which our defined proxies
for physical parameters could be measured from these datasets. Under April
weather conditions and using an observing frequency of 230~GHz, we predict that
a ``Phase 1'' ngEHT can potentially measure $\sim$50 black hole masses,
$\sim$30 black hole spins, and $\sim$7 black hole shadows across the entire
sky.
