Couldn’t the Westerlands and the Vale and other mountainous regions grow potatoes? (I thought they had potatoes in Westeros.)

Nope. At least in the books, potatoes do not exist in Westeros, nor do any other “New World” flora. 

The Vale does grow a lot of food – but that’s in the Vale (i.e, “the val(l)e(y)”), not up in the mountains.