We present an analysis of the JWST NIRCam and MIRI morphological properties of 80 massive (log₁₀(M_[M_])=11.2±0.1) dusty star-forming galaxies at z=$2.7$+1.2-0.7$, identified as sub-millimetre galaxies (SMGs) by ALMA, that have been observed as part of the JWST PRIMER project. To compare the structure of these massive, active galaxies to more typical star-forming galaxies, we define a sample of 850 field galaxies with matched redshifts and specific star formation rates. We identify 20±5% of the SMGs as candidate late-stage major mergers, a further 40±10% as potential minor mergers and 40±10% which have comparatively undisturbed disk-like morphologies, with no obvious massive neighbours on 20-30kpc (projected) scales. These rates are comparable to those for the field sample and indicate that the majority of the sub-millimetre-detected galaxies are not late-stage major mergers, but have interaction rates similar to the less-active population at z~$2-3. Through a multi-wavelength morphological analysis, we establish that SMGs have comparable near-infrared sizes to the less active population, but exhibit lower S\'ersic indices, consistent with bulge-less disks and have more structured morphologies at 2μm relative to 4μm. We find evidence for dust reddening as the origin of the morphological differences between the populations, identifying a strong correlation between the F200W$-$F444W pixel colour and the 870μm surface brightness. We conclude that SMGs and less active galaxies at the same epochs share a common disk-like structure, but the weaker bulge components of the SMGs results in a lower dynamical stability. Consequently, instabilities triggered either secularly or by minor external perturbations result in higher levels of activity (and dust content) in SMGs compared to typical star-forming galaxies. [Abridged]
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