You can actually compute that! Model your agent’s dynamics (even just pick a double integrator, I.e., an agent where the acceleration is controlled, or something) and pick a value for the GPS covariance. You can compute the infinite-time limit of the variance of the Kalman filter exactly (your favorite numerical package has a function for solving algebraic Riccati equations which is all you need to do this). No simulated observations are necessary, just the dynamics and the GPS noise.

My guess is that you’ll find the filter converges to a distribution that has a decently large covariance if the military didn’t want other forces using it to make good decisions.