In the absence of data from a randomized trial, researchers may aim to use observational data to draw causal inference about the effect of a treatment on a time-to-event outcome. In this context, interest often focuses on the treatment-specific survival curves, that is, the survival curves were the population under study to be assigned to receive the treatment or not. Under certain conditions, including that all confounders of the treatment-outcome relationship are observed, the treatment-specific survival curve can be identified with a covariate-adjusted survival curve. In this article, we propose a novel cross-fitted doubly-robust estimator that incorporates data-adaptive (e.g. machine learning) estimators of the conditional survival functions. We establish conditions on the nuisance estimators under which our estimator is consistent and asymptotically linear, both pointwise and uniformly in time. We also propose a novel ensemble learner for combining multiple candidate estimators of the conditional survival estimators. Notably, our methods and results accommodate events occurring in discrete or continuous time, or an arbitrary mix of the two. We investigate the practical performance of our methods using numerical studies and an application to the effect of a surgical treatment to prevent metastases of parotid carcinoma on mortality.