I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium.
But the charge against those who are "deeply attached to the mathematical idealism of eigenvalue-based solutions" has an edge to it which could easily be taken out of context.
Your point is that the 'first order' L-V model, which doesn't take into account forcing, won't give accurate predictions in real-world scenarios. Fine.
But eigenvalue-based solutions and equilibria are absolutely foundational concepts, which apply almost perfectly in cases like chemical reactions, which are not dominated by external forces. So let's not get discouraged from opening up the topic just because it's empirical applicability needs to be evaluated on a case-by-case basis.
I can also imagine that even where it doesn't literally apply, the theory of equilibrium may provide a framework for understanding the disruption of equilibrium by slight to moderate amounts of forcing. For example, with slight forcing, there will no longer be points of equilibrium, but perhaps small fuzzy regions of "quasi-equilibrium" around the theoretical equilibrium points.