  Magnetic topological insulators (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$
($n=0,1,2,3$) are promising to realize exotic topological states such as the
quantum anomalous Hall effect (QAHE) and axion insulator (AI), where the
Bi$_2$Te$_3$ layer introduces versatility to engineer electronic and magnetic
properties. However, whether surface states on the Bi$_2$Te$_3$ terminated
facet are gapless or gapped is debated, and its consequences in thin-film
properties are rarely discussed. In this work, we find that the Bi$_2$Te$_3$
terminated facets are gapless for $n \ge 1$ compounds by calculations. Despite
that the surface Bi$_2$Te$_3$ (one layer or more) and underlying MnBi$_2$Te$_4$
layers hybridize and give rise to a gap, such a hybridization gap overlaps with
bulk valence bands, leading to a gapless surface after all. Such a metallic
surface poses a fundamental challenge to realize QAHE or AI, which requires an
insulating gap in thin films with at least one Bi$_2$Te$_3$ surface. In theory,
the insulating phase can still be realized in a film if both surfaces are
MnBi$_2$Te$_4$ layers. Otherwise, it requires that the film thickness is less
than 10$\sim$20 nm to push down bulk valence bands via the size effect. Our
work paves the way to understand surface states and design bulk-insulating
quantum devices in magnetic topological materials.
