We investigate the radiative stability of Horndeski scalar-tensor theories with luminally propagating gravitational waves (as extensively discussed in the wake of GW170817) and show that in general there is a tension between obtaining observable deviations from general relativity (GR) in cosmology and the requirement of radiative stability. Using this as a constraint, we discuss the subsets of theories that are capable of evading this conclusion and yielding observable, radiatively stable departures from GR. We find several classes of theories that can do so, recovering known cases and identifying several additional radiatively stable cases. Finally we also extract the cosmological signatures of two particularly well-motivated radiatively stable classes of theories: shift-symmetric theories and theories with a conformal coupling between the scalar and gravity. We find that cosmological parameter constraints on dark energy and modified gravity parameters for both of these two classes, which we explicitly compute using data from the Planck, SDSS/BOSS and 6dF surveys, are significantly tightened with respect to generic Horndeski theories.