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While some of the most elegant applications of topological insulators, such as quantum anomalous Hall effect, require the preservation of Dirac surface states in the presence of time-reversal symmetry breaking, other phenomena such as spin-charge conversion rather rely on the ability for these surface states to imprint their spin texture on adjacent magnetic layers. In this work, we investigate the spin-momentum locking of the surface states of a wide range of monolayer transition metals (3$d$-TM) deposited on top of Bi$_{2}$Se$_{3}$ topological insulators using first principles calculations. We find an anticorrelation between the magnetic moment of the 3$d$-TM and the magnitude of the spin-momentum locking {\em induced} by the Dirac surface states. While the magnetic moment is large in the first half of the 3$d$ series, following Hund's rule, the spin-momentum locking is maximum in the second half of the series. We explain this trend as arising from a compromise between intra-atomic magnetic exchange and covalent bonding between the 3$d$-TM overlayer and the Dirac surface states. As a result, while Cr and Mn overlayers can be used successfully for the observation of quantum anomalous Hall effect or the realization of axion insulators, Co and Ni are substantially more efficient for spin-charge conversion effects, e.g. spin-orbit torque and charge pumping.