Bias correction : Why use a linear model?

In the section for Bias correction a linear regression model is used to find u(X), specifically it is Y ~ X. My question is :

  1. Why should it be a linear model and not some higher degree (Y ~ X^2 + X), or, perhaps a neural net?
  2. In case of multiple covariates, should interaction terms be also included?

I do not have answers your questions yet but am trying to figure it out as I read this chapter!

So I did ask a colleague who attended a session from Scott Cunningham and he says that it could be any kind of model.

Firstlly, invite your colleague to attend! Secondly thanks for asking, I wonder how the choice of model here is justified in real life then. in the text its just stated we will use OLS, I wonder if this is just commonly accepted or if there are common alternatives

It can be any model. OLS is the more traditional/conventional way. I believe, since these causal inference techniques originated mostly from econometrics and not, say, AI practitioners, OLS has traditionally been used.