In this paper, we revisit the problem of identifying bona fide cluster Cepheids by performing an all-sky search for Cepheids associated with open clusters and making use of state-of-the-art catalogued information for both Cepheids and clusters, based on the unparalleled astrometric precision of the second and early third data releases of the Gaia satellite. We determine membership probabilities by following a Bayesian approach using spatial and kinematic information of the potential cluster-Cepheid pairs. We confirm 19 Cepheid-cluster associations considered in previous studies as bona fide, and question the established cluster membership of six other associations. In addition, we identify 138 cluster Cepheid candidates of potential interest, mostly in recently discovered open clusters. We report on at least two new clusters possibly hosting more than one Cepheid. Furthermore, we explore the feasibility of using open clusters hosting Cepheids to empirically determine the Cepheid period-age relation through the use of Gaia and 2MASS photometry and a semiautomated method to derive cluster ages. We conclude that the usage of cluster Cepheids as tentative probes of the period-age relations still faces difficulties due to the sparsely populated red giant branch and the stochastically sampled main-sequence turn-off of the open clusters, making age determinations a challenging task. This biases the age-datable cluster selection for Cepheid period-age studies towards older and high-mass clusters.