This crossed my mind at one point.
It's not a bad idea, ubt I would like to deepen it a bit.
My initial thought was what you said. Some mutations would make you unable to wear certain headwear. But I would also like to add something similiar to outfits. Wearing plate armor with back spines and wings would be inpossible for example. To counter that, I would also like to add some mutat exclusive armor. Helmets that actualy require you to have horns, armor that can only be worn if you have tail and so on.
On top of that, you will be able to get rid of mutations by surgery and gene editing in the future, but I would also like to add a simple, less sophisticated way of removing them. Cutting your horns and filing them down, amputating your tail and so on. There would be disadvantages to this, but it makes sence to be able to do this.
All of this will require quite a bit of time though. I will probably implement only small part of what I want first and add the rest later. We will see.