There are a fair number of extensions to GR that can reproduce the standard cosmology, typically avoiding or recasting DM, DE, or both. Examples include generalized teleparallel gravity and Cotton gravity. In such theories, any solution of the EFEs are solutions of the (sometimes very different) field equations of these families of theories, although the field content may have a somewhat different physical interpretation from GR.
However, generically, these extensions tend to have an under-determination problem frustrating attempts to arrive at a unique spacetime for a given distribution of matter or a unique distribution of matter which can exist in a given spacetime (or both problems). That makes them less attractive than GR, or even possibly outright unsuitable bases for initial-values formulations (and thus are unlikely to overthrow numerical relativity soon).
However, generically, these extensions tend to have an under-determination problem frustrating attempts to arrive at a unique spacetime for a given distribution of matter or a unique distribution of matter which can exist in a given spacetime (or both problems). That makes them less attractive than GR, or even possibly outright unsuitable bases for initial-values formulations (and thus are unlikely to overthrow numerical relativity soon).