dN/dt = kN says a population's growth is proportional to its size, and predicts unbounded exponential growth. Why does that model still earn its place given no population does that?

B. Because it is accurate while resources are unlimited

It is right for the early phase and wrong later, and the shape of its failure names the missing ingredient, a carrying capacity, which the logistic equation adds. A simple model that fails informatively is often more useful than a complicated one that fits, because you can see what it left out.