We investigate the mechanisms underlying the generation of broadband trailing-edge noise from an airfoil, focusing on the role of hydrodynamic structures with nonzero spanwise wavenumbers and the development of a corresponding reduced-order model. A wall-resolved compressible large-eddy simulation (LES) of a NACA0012 airfoil at 3 degrees angle of attack, Reynolds number Re = 200 000, and Mach number M = 0.3 is performed. The simulation replicates the experimental setup of Demange et al. [AIAA Aviation Forum (2023)], including the tripping elements ensuring turbulent boundary layer development. The numerical domain has a large spanwise extent (43.75% of the chord) to allow for the investigation of large spanwise coherent structures. Validation against experimental data shows good agreement. The simulation results show that acoustic radiation from the trailing edge follows the scattering condition, whereby only hydrodynamic waves with spanwise wavenumbers lower than the acoustic wavenumber can generate propagative sound. Notably, the contribution from nonzero spanwise wavenumber modes becomes significant beyond the cut-on frequency at which the first nonzero spanwise acoustic mode becomes propagative. Through spectral proper orthogonal decomposition (SPOD), the associated flow structures are identified as streamwise-travelling wave packets convecting obliquely across the trailing edge. Reduced-order models based on acoustic-domain extended SPOD accurately reproduce the LES sound pressure levels within 2 dB, using only two modes per frequency-wavenumber pair. This compact representation paves the way for targeted noise-reduction strategies based on the control of the identified coherent structures.