So I know that Einstein and general relativity had huge impacts on the way we view the world, but how crucial were these scientific advancements to the success of our space programs? Would Newtonian physics sufficed, or would using those formulas and methodologies when travelling off-Earth and to the Moon resulted in catastrophic failures?
Well, relativistic effects are accounted for in all space travel. These effects were apparently minor enough during the Apollo missions to rely on standard Newtonian equations and minor course corrections enroute, but are pronounced enough on longer missions within the solar system to account for several kilometer errors.
General relativity is routinely accounted for in spacecraft navigation.... the NASA navigation software developed at JPL....incorporates the Ted Moyer formulations for navigation, which includes mathematical expressions that describe the effects of general relativity.