Recent JWST observations of the Type Ia supernova (SN Ia) 2021aefx in the nebular phase have paved the way for late-time studies covering the full optical to mid-infrared (MIR) wavelength range, and with it the hope to better constrain SN Ia explosion mechanisms. We investigate whether public SN Ia models covering a broad range of progenitor scenarios and explosion mechanisms can reproduce the full optical-MIR spectrum of SN 2021aefx at $\sim$270 days post explosion. We perform 1D steady-state non-LTE simulations with the radiative-transfer code CMFGEN, and compare the predicted spectra to SN 2021aefx. The models can explain the main features of SN 2021aefx over the full wavelength range. However, no single model, or mechanism, emerges as a preferred match. We discuss possible causes for the mismatch of the models, including ejecta asymmetries and ionisation effects. Our new calculations of the collisional strengths for Ni III have a major impact on the two prominent lines at 7.35 and 11.00 $\mu$m, and highlight the need for more accurate collisional data for forbidden transitions. Using updated atomic data, we identify a strong feature due to [Ca IV] 3.21 $\mu$m, attributed to [Ni I] in previous studies. We also provide a tentative identification of a forbidden line due to [Ne II] 12.81 $\mu$m, whose peaked profile suggests that neon is mixed inwards during the explosion, as predicted for instance in violent merger models. Contrary to previous claims, we show that the [Ar III] 8.99 $\mu$m line can be broader in sub-$M_\mathrm{Ch}$ models compared to near-$M_\mathrm{Ch}$ models. Our models suggest that key physical ingredients are missing from either the explosion models, or the radiative-transfer post-processing, or both. Nonetheless, they also show the potential of the near- and mid-infrared to uncover new spectroscopic diagnostics of SN Ia explosion mechanisms. [Abridged]