  Upcoming measurements of the small-scale primary cosmic microwave background
(CMB) temperature and polarization power spectra ($TT$/$TE$/$EE$) are
anticipated to yield transformative constraints on new physics, including the
effective number of relativistic species in the early universe ($N_{\rm eff}$).
However, at multipoles $\ell \gtrsim 3000$, the primary CMB power spectra
receive significant contributions from gravitational lensing. While these modes
still carry primordial information, their theoretical modeling requires
knowledge of the CMB lensing convergence power spectrum, $C_L^{\kappa\kappa}$,
including on small scales where it is affected by nonlinear gravitational
evolution and baryonic feedback processes. Thus, the high-$\ell$ primary CMB is
sensitive to these late-time, nonlinear effects. Here, we show that
inaccuracies in the modeling of $C_L^{\kappa\kappa}$ can yield surprisingly
large biases on cosmological parameters inferred from the primary CMB power
spectra measured by the upcoming Simons Observatory and CMB-S4 experiments. For
CMB-S4, the biases can be as large as $1.6\sigma$ on the Hubble constant $H_0$
in a fit to $\Lambda$CDM and $1.2\sigma$ on $N_{\rm eff}$ in a fit to
$\Lambda$CDM+$N_{\rm eff}$.
  We show that these biases can be mitigated by explicitly discarding all $TT$
data at $\ell>3000$ or by marginalizing over parameters describing baryonic
feedback processes, both at the cost of slightly larger error bars. We also
discuss an alternative, data-driven mitigation strategy based on delensing the
CMB $T$ and $E$-mode maps. Finally, we show that analyses of upcoming data will
require Einstein-Boltzmann codes to be run with much higher numerical precision
settings than is currently standard, so as to avoid similar -- or larger --
parameter biases due to inaccurate theoretical predictions.
