We investigate the formation of flat bands in a two-dimensional electron gas patterned by a periodic triangular array of repulsive antidots. In the high-barrier, narrow-trench regime, the spectrum develops kagome-like band triplets, each comprising a nearly flat band and two dispersive bands, with wave functions concentrated along the potential-free trenches. The two lowest energy bands have a different character with graphene-like dispersion and wave functions localized near the junction centers. The appearance of the trench-concentrated nearly flat bands further motivates a honeycomb quantum graph model as an effective description of the narrow-trench limit. In the single-channel graph, destructive interference produces exact flat bands with compact localized eigenstates protected by the D₃ symmetry. Their energies obey a quantization condition determined by propagation along the links and scattering at the junctions. We then extend the analysis to a multichannel quantum graph and derive the most general junction scattering matrix compatible with D₃ symmetry. Even- and odd-parity transverse channels contain equivalent symmetry sectors, and D₃ symmetry therefore allows them to mix. This mode conversion can disrupt the destructive interference responsible for flat bands and generically gives those bands a finite width. Exact flatness is nevertheless retained when only one transverse-parity sector is active, or when an additional symmetry forbids mixing between the two sectors. These results connect the continuum antidot model to its quantum-graph limit and identify transverse-mode mixing as a mechanism that limits flatness in multichannel networks.