How biological networks achieve robust control despite relying on imperfect, local information remains an important open question. Here, we identify thermodynamic constraints that can curtail non-equilibrium steady-state responses so severely that even crude, local feedback rules can achieve globally stable control without requiring precise network design or global information. Specifically, using Markov jump processes as a general framework for biophysical dynamics, we derive general non-equilibrium response constraints showing that for many classes of rate perturbations, steady-state responses have fixed signs across all driving strengths, so that near-equilibrium responses predict far-from-equilibrium behavior regardless of system complexity. These constraints clarify several biological phenomena: monotonicity is thermodynamically guaranteed whenever a perturbation acts on a single transition rate, and non-monotonic responses, as observed for example in transcription factor regulation, arise only when an input simultaneously modulates multiple rates. Even in this case, we identify a graph-theoretic concept termed "coherence" that allows for a restoration of monotonicity. We show how coherence naturally and generally emerges in classic biophysical models of adaptation, including chemotaxis, and transcription factor regulation when biological constraints on network parameterization are included. We next show that, within a control-theoretic framework, these constraints guarantee that simple linear feedback on small subsets of kinetic rates achieves globally stable tracking and adaptation without coordinated manipulation of many variables. For systems with one regulator, local stability implies global stability for arbitrary network topologies without fine tuning. For multiple regulators, the thermodynamic constraints restrict the allowed nullcline geometry in ways that preclude unstable dynamical structures. We also show that stable control with non-monotonic responses can be achieved by tuning the signs of different feedback couplings for the regulated variable. Together, these results reveal that non-equilibrium thermodynamics fundamentally constrains biochemical network responses, explaining when and why responses are monotonic or non-monotonic and guaranteeing the effectiveness of minimal, local feedback for robust control in complex reaction networks.