That is exactly the point! The Standard Model does not make a prediction for the bare Higgs mass which needs its fine-tuned value in this model.This is not quite a type of example I was looking for. There was no "deferent and epicycle" atomic theory which was predicting atomic energy levels, before we've got our current one.
An easy explanation for something that looked mysterious before. In other words, a good theory.But the explanation that evolution can offer here is not any sophisticated mechanism, but simply selection bias.