Thermal enhancement in household, biomedical, and industrial appliances remains essential for technological advancement, thus motivating the search for advanced heat-transfer fluids with superior thermal and electromagnetic performance. Although hybrid nanofluids have shown considerable promise, the specific role of lightweight and corrosion-resistant aluminum-alloy nanoparticles (AA7072 and AA7075) dispersed in ethylene glycol under the combined influences of Hall current, ion-slip effect, and Darcy–Forchheimer permeability has not been previously reported. The present work addresses this gap by developing a three-dimensional, steady, incompressible magnetohydrodynamic (MHD) model for the flow of a Casson hybrid nanofluid over a vertically stretching sheet, incorporating nonlinear thermal radiation, viscous dissipation, Joule (Ohmic) heating, and heat generation. The thermophysical properties of the hybrid nanofluid are modeled within the Tiwari–Das framework, while the generalized Ohm’s law with Hall and ion-slip terms governs the electromagnetic body forces. Key assumptions include thermal equilibrium between the nanoparticles and the base fluid, the Boussinesq approximation for buoyancy, and the Rosseland approximation for radiative heat flux. The governing partial differential equations are reduced to systems of nonlinear ordinary differential equations via appropriate similarity transformations and solved numerically using the Chebyshev Collocation Method (CCM), whose accuracy and efficiency are validated against published benchmarks. The analysis reveals that: increasing the Hall parameter diminishes the Lorentz force by reducing effective electrical conductivity, thereby augmenting primary fluid velocity; the ion-slip parameter further boosts velocity through enhanced charge mobility while simultaneously suppressing the temperature field via reduced Joule heating; higher values of the nonlinear thermal radiation parameter amplify the thermal profile.