We present a comprehensive study of the physical origin of radio emission in optical quasars at redshifts $z<2.5$. We focus particularly on the associations between compact radio emission, dust reddening, and outflows identified in our earlier work. Leveraging the deepest low-frequency radio data available to date (LoTSS Deep DR1), we achieve radio detection fractions of up to 94<!PCT!>, demonstrating the virtual ubiquity of radio emission in quasars, and a continuous distribution in radio loudness. Through our analysis of radio properties, combined with spectral energy distribution modelling of deep multiwavelength photometry, we establish that the primary source of radio emission in quasars is the active galactic nucleus (AGN), rather than star formation. Modelling the dust reddening of the accretion disc emission shows a continuous increase in radio detection in quasars as a function of the reddening parameter E(B-V) suggesting a causal link between radio emission and dust reddening. Confirming previous findings, we observe that the radio excess in red quasars is most pronounced for sources with compact radio morphologies and intermediate radio loudness. We find a significant increase in Oiii and Civ outflow velocities for red quasars not seen in our control sample, with particularly powerful Oiii winds in those around the threshold from radio-quiet to radio-loud. Based on the combined characterisation of radio, reddening, and outflow properties in our sample, we favour a model in which the compact radio emission observed in quasars originates in compact radio jets and their interaction with a dusty, circumnuclear environment. In particular, our results align with the theory that jet-induced winds and shocks resulting from this interaction are the origin of the enhanced radio emission in red quasars. Further investigation of this model is crucial for advancing our understanding of quasar feedback mechanisms and their role in galaxy evolution.