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We present an analysis of the quenching of star formation in galaxies, bulges, and disks throughout the bulk of cosmic history, from $z=2-0$. We utilise observations from the SDSS and MaNGA at low redshifts. We complement these data with observations from CANDELS at high redshifts. Additionally, we compare the observations to detailed predictions from the LGalaxies semi-analytic model. To analyse the data, we developed a machine learning approach utilising a Random Forest classifier. We first demonstrate that this technique is extremely effective at extracting causal insight from highly complex and inter-correlated model data, before applying it to various observational surveys. Our primary observational results are as follows: At all redshifts studied in this work, we find bulge mass to be the most predictive parameter of quenching, out of the photometric parameter set (incorporating bulge mass, disk mass, total stellar mass, and $B/T$ structure). Moreover, we also find bulge mass to be the most predictive parameter of quenching in both bulge and disk structures, treated separately. Hence, intrinsic galaxy quenching must be due to a stable mechanism operating over cosmic time, and the same quenching mechanism must be effective in both bulge and disk regions. Despite the success of bulge mass in predicting quenching, we find that central velocity dispersion is even more predictive (when available in spectroscopic data sets). In comparison to the LGalaxies model, we find that all of these observational results may be consistently explained through quenching via preventative `radio-mode' active galactic nucleus (AGN) feedback. Furthermore, many alternative quenching mechanisms (including virial shocks, supernova feedback, and morphological stabilisation) are found to be inconsistent with our observational results and those from the literature.