  The Epoch of Reionization (EoR) depends on the complex astrophysics governing
the birth and evolution of the first galaxies and structures in the
intergalactic medium. EoR models rely on cosmic microwave background (CMB)
observations, and in particular the large-scale E-mode polarization power
spectra (EE PS), to help constrain their highly uncertain parameters. However,
rather than directly forward-modelling the EE PS, most EoR models are
constrained using a summary statistic -- the Thompson scattering optical depth,
$\tau_e$. Compressing CMB observations to $\tau_e$ requires adopting a basis
set for the EoR history. The common choice is the unphysical,
redshift-symmetric hyperbolic tangent (Tanh) function, which differs in shape
from physical EoR models based on hierarchical structure formation. Combining
public EoR and CMB codes, 21cmFAST and CLASS, here we quantify how inference
using the $\tau_e$ summary statistic impacts the resulting constraints on
galaxy properties and EoR histories. Using the last Planck 2018 data release,
we show that the marginalized constraints on the EoR history are more sensitive
to the choice of the basis set (Tanh vs physical model) than to the CMB
likelihood statistic ($\tau_e$ vs PS). For example, EoR histories implied by
the growth of structure show a small tail of partial reionization extending to
higher redshifts. However, biases in inference using $\tau_e$ are negligible
for the Planck 2018 data. Using EoR constraints from high-redshift observations
including the quasar dark fraction, galaxy UV luminosity functions and CMB EE
PS, our physical model recovers $\tau_e=0.0569^{+0.0081}_{-0.0066}$.
