In the context of the effective field theory of dark energy (EFT) we perform agnostic explorations of Horndeski gravity. We choose two parametrizations for the free EFT functions, namely, a power law and a dark energy density-like behavior on a nontrivial Chevallier-Polarski-Linder background. We restrict our analysis to those EFT functions which do not modify the speed of propagation of gravitational waves. Among those, we prove that one specific function cannot be constrained by data since its contribution to the observables is below the cosmic variance, although we show it has a relevant role in defining the viable parameter space. We place constraints on the parameters of these models by combining measurements from present-day cosmological data sets, and we prove that the next-generation galaxy surveys can improve such constraints by 1 order of magnitude. We then prove the validity of the quasistatic limit within the sound horizon of the dark field, by looking at the phenomenological functions μ and Σ , associated, respectively, with clustering and lensing potentials. Furthermore, we notice up to 5% deviations in μ , Σ with respect to general relativity at scales smaller than the Compton one. For the chosen parametrizations and in the quasistatic limit, future constraints on μ and Σ can reach the 1% level and will allow us to discriminate between certain models at more than 3 σ , provided the present best-fit values remain.