Understanding the origin and evolution of carbon-enhanced metal-poor (CEMP) stars is key to tracing the early chemical enrichment of the Galaxy. In this work, we investigate how physically realistic 3D radiation-hydrodynamics (RHD), carbon-enhanced model atmospheres affect the inferred carbon abundances in CEMP stars, and assess the implications for their classification and for the Galactic chemical evolution (GCE). We pay particular attention to the systematic biases introduced by traditional 1D hydrostatic equilibrium (HE) models. We use the M3DIS code to compute 3D RHD model atmospheres for main-sequence and sub-giant stars spanning a wide range of metallicities and carbon enhancements. Synthetic spectra of the CH G-band are calculated using full 3D radiative transfer, and compared to spectra from classical 1D HE MARCS models. We derive abundance corrections and apply them to a large literature sample of metal-poor stars from the SAGA database to quantify systematic effects on the carbon abundance distribution and CEMP classification. Our new 3D CEMP models predict significantly cooler upper atmospheric layers than present in 1D HE models, resulting in stronger CH absorption and lower inferred carbon abundances by up to -0.9dex at the lowest metallicities. Carbon enhancement in the atmosphere itself increases molecular opacities and leads to radiative re-heating, which partly offsets the adiabatic cooling in 3D models and reduces the magnitude of 3D-1D abundance corrections. Applying these corrections lowers the CEMP fraction by up to 20% below [Fe/H]=-3, and furthermore alters the relative contribution of CEMP sub-classes. In particular, the fraction of CEMP-no stars increases while the number of stars classified as CEMP-r/s decreases, owing to the downward revision of absolute carbon abundances. These changes bring the Galactic carbon abundance distribution into better agreement with GCE models that assume a contribution from faint supernovae of 20%. Physically realistic model atmospheres are thus essential for reliable reconstruction of the early chemical enrichment history of the Galaxy.