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We estimate disc-averaged RV variations of the Sun over the last magnetic cycle, from the single Fe I line observed by SDO/HMI, using a physical model for rotationally modulated magnetic activity that was previously validated against HARPS-N solar observations. We estimate the disc-averaged, unsigned magnetic flux and show that a simple linear fit to it reduces the RMS of RV variations by 62%, i.e. a factor of 2.6. We additionally apply the FF' method, which predicts RV variations based on a star's photometric variations. At cycle maximum, we find that additional physical processes must be at play beyond suppression of convective blueshift and velocity imablances resulting from brightness inhomogeneities, in agreement with recent studies of solar RV variations. By modelling RV variations over the magnetic cycle using a linear fit to the unsigned magnetic flux, we recover injected planets at an orbital period of about 300 days with RV semi-amplitudes down to 0.3 m/s. To reach semi-amplitudes of 0.1 m/s, we will need to identify and model additional physical phenomena that are not well traced by the unsigned magnetic flux or FF'. The unsigned magnetic flux is an excellent proxy for rotationally modulated, activity-induced RV variations, and could become a key tool in confirming and characterising Earth analogs orbiting Sun-like stars. The present study motivates ongoing and future efforts to develop observation and analysis techniques to measure the unsigned magnetic flux at high precision in slowly rotating, relatively inactive stars like the Sun.