I think what you're saying is orthogonal to what I'm saying.
Yes, ZIP codes can be a proxy for race in models dealing with credit scores (and recidivism, which is also a really bad place to put racial bias), as an example. But if I put it in a mixed-effects model, it shows the same bias, and a mixed-effects model is just an extended version of linear regression. Both statistical and ML models suffer from the problem you're stating.
What you have not made any assumptions about in a random forest is about the distribution of the data you're looking at. One example of a case where the assumptions that bog-standard OLS makes about your data can cause you problems is zero-dominated data -- data with a lot of zeros in it. Basically any time you're trying to make predictions about things that are rare in your measured population.
OLS does a bad job on zero-dominated data. If you throw a zero-dominated dataset into a random forest, you will get back better answers than if you use OLS on zero-dominated data.
To be clear: there are strategies for dealing with zero-dominated data using statistical inference. You don't have to resort to non-inferential learning just because you have data that doesn't look like a bell curve. But machine learning is a powerful way to get pretty good results on a lot of problem spaces without having to understand the probability function involved (or in cases where the probability function is too complicated to be tractable computationally, like the probability function that determines the color of pixels in a dataset where you're classifying dogs versus cats).