Study of the KPZ universality class has seen the emergence of universal objects over the past decade which arise as the scaling limit of the member models. One such object is the directed landscape, and it is known that exactly solvable last passage percolation (LPP) models converge to the directed landscape under the KPZ scaling (see \cite{DV21}). Large deviations of the directed landscape on the metric level were recently studied in \cite{DDV24}, which also provides a general framework for establishing such large deviation principle (LDP). The main goal of the article is to apply and refine that framework to establish a LDP for LPP models at the metric level without relying on exact solvability. We then use the LDP on the metric level to establish a LDP for geodesics in these models, thus providing a streamlined way to study large transversal fluctuations of geodesics in these models. We briefly touch on how the theory extends to other planar models like directed polymers and Poisson LPP.