Most do chit indoors, however it is likely a case of convenience and you can see them easily. Outside may at present be a little early owing to frosts.
Not personally sure about chitting. I never have bothered. There was some years ago a program about Jersey Royals, or Jersey or the Channel islands.
They chitted the seed potatoes in large frames for the purpose, but they then rubbed out the first ones that appeared. The principal was that if they initially formed 2 chits and these were rubbed out then the seed potato "panicked" and then threw out 3, or maybe more, chits. This therefore gave them more stems on which potatoes grew and a bigger crop.
The aspect I can see is that we as gardeners chit, get 2 chits or sprouts and simply plant the 2 sprout potato. So we are likely not making any difference.
The other thought is that if we plant a potato with 3 good chits then in planting and covering there is a fair chance we could cause a chit to break and so reduce the potato bearing stems that grow.
Back to "damage" probably none. They are after all from Peru and from the Andes which gets cold. If the whole potato froze then likely murder, otherwise I would expect little.