We study a random walk driven by a particle system from a generic class, and establish a law of large numbers for the walk for almost all densities of the environment. To do so, we exploit the finite-ranged approximations of the environment from arXiv:2409.02096 in a new way, whereby the monotonicity (in the density) of the walker's displacement is leveraged to show the existence of an actual speed. This bypasses the constructions in arXiv:1906.03167 and generalises its Theorem 1.1, which applied to specific environments. We illustrate this with a family of particle systems where the particles have underlying drifts, namely mixtures of APCRWs (Asymmetric Poisson Cloud of Random Walks). In particular, when all particles have the same drift, we prove the LLN under any choice of parameters save one critical density of the environment. To our knowledge, this is the first time that such a conservative and slow-mixing environment with drift is treated outside of the non-nestling case (in which the walker is already assumed to travel strictly faster/slower than the drift arXiv:2205.00282).