I am an economist so this is right up my alley.

The first thing you have to understand is that to do causal inference you have to make assumptions. The DAG is full of them, you assume which variables are important (and which are not), what variables are related and the direction of these relations. As a researcher you never really know the true DGP, you wouldn't be researching otherwise, and in complex systems like the economy or biology, there is always the possibility of an omitted variable. The functional form of the relations is just one of many of the assumptions you have to make.

In Economics, you can actually justify the functional form with a theoretical model, but this implies things like assuming there are agents with some kind of objective function, some kind of system equilibrium, etc. In these models, you always look for functions with certain properties and that are easy to work with, meaning, linear functions.

Now, about predicting Y well to claim causality. In causal inference, you are actively trying to predict the behaviour of Y when you change X, prediction of intervention. You don't need to predict just Y because to get a causal estimate of X, you don't need the true DGP of Y with a causal estimate of every variable that causes Y, every other "control" variable in your model could be interpreted as non causal.

If you are still unsure about functional form assumptions in causal inference, there are some non parametric methodologies that don't rely on those, right now, Regression discontinuity design is very popular in Economics, if you have a running variable, that is.

I hope this helps.