I think there is usually some transform that can convert your problem into a linear, or, in general, easy problem.

I don't think this is true. The model must reflect the underlying reality and the underlying reality just isn't reliably linear, even after transforms.

Now, historically people used to greatly prefer linear models. Why? Because they were tractable. And for something that you couldn't convert into linear, well, you just weren't able to build a good model. However in our computer age this no longer holds.

For an example consider what nowadays is called "machine learning". They are still building models, but these tend to be highly non-linear models with no viable linear transformations.