Econometrics is a separate discipline from statistics/probability/machine learning for a reason. The primary aim of econometrics and economics in general is interpretability. Economists could easily build very realistic models with hundreds of moving parts/variables, but they don’t do it, because they aim to make models as simple as possible.
In this sense, econometrics does not pursue precision, it serves as the best demonstration of economic theory. It doesn’t answer the question “How close?”, it answers the question “Why?”
Actually, econometrics does use nonlinear models a lot, but it always remarks that these models are “atheoretical”, i.e. not based on economic theory, which is the cornerstone of econometrics - to serve economic theory.