This letter introduces HALO-LP, a novel homogeneous all-shifted non-interior-point method for linear programming (LP). The proposed method combines a homogeneous self-dual (HSD) embedding with all-shifted nonlinear complementarity (NCP) functions and proximal primal-dual regularization, enabling arbitrary primal-dual warm starts without requiring strictly interior iterates. The resulting Newton equations admit a condensation based on a positive-definite matrix that can be efficiently factorized using Cholesky decomposition, while the HSD embedding enables primal and dual infeasibility detection through Farkas certificates. Numerical evaluations on perturbed LP instances derived from the Netlib test set demonstrate state-of-the-art warm-starting performance, outperforming the competing solvers in the evaluated setting. We further validate optimality and infeasibility detection on analytically characterized LP problems. Finally, we demonstrate HALO-LP's applicability to robotic optimal control through sequential linear programming (SLP) for centroidal walking and trotting of quadrupedal robots.