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, $C_L^{kappakappa}$, 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 $C_L^{kappakappa}$ 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 $1.6sigma$ on the Hubble constant $H_0$ in a fit to $Lambda$CDM and $1.2sigma$ 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.
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