For what it's worth from someone who has made a couple...
Most cheap gunstocks are made using whatever is the cheapest method, i think copy routing is rather usual, and probably CNC too.
However, for nicer stocks, they are usually made by hand.
There's a few pitfalls to making your own, but that is primarily the geometry of the butt of the stock, something which is explained in great detail on several sites on the big interweb and the recoil forces which will be working on the stock.
The main things to consider IMHO is the wood in relation to recoil forces (Not such a big deal on smaller calibre rifles, but a VERY big deal on larger calibers, and it does have an impact especially on how well the stock will hold up to use), layout of the stock, and any special needs which might need tending to in relation to the person who will be using it.
If it's your first rifle stock, i suggest a nice plank of beech or birch to try out on, before making it in nicer wood like walnut. Stock blanks is readily available in the UK AFAIK.