Plasma streaming instabilities excited by relativistic charged particle beams play a pivotal role in astrophysical and laboratory environments. Their numerical study, however, is challenged by the disparity in spatiotemporal scales between the background plasma and beam particles, which can differ by several orders of magnitude for tenuous, ultrarelativistic beams. Here, we exploit the quasistatic approximation (QSA) to develop a new theoretical framework capable of capturing the full unstable spectrum in the spatiotemporal regime relevant for beams that continuously encounter unperturbed plasma at their leading edge. Within this linear, fully electromagnetic model, we uncover a previously unreported spatiotemporal evolution of the current filamentation instability and elucidate its interplay with the oblique two-stream instability, predicting the dominance of filamentation in the vicinity of the beam front. The good agreement between theory, kinetic particle-in-cell (PIC) simulations, and QSA-based PIC simulations validates the robustness of the approach. By pushing QSA-based PIC simulations to extremely dilute electron-positron beams, such as those found in blazar jets, we demonstrate their unique ability to capture the rich nonlinear dynamics of streaming instabilities in parameter regimes previously inaccessible to kinetic simulations.