Because that would call action_index(), and that doesn't exist?
And yes, you must add the "action_" prefix to every method that is routeable. Not only because that allows you to create non-routeable public methods too (methods that you want to call but are not output generating methods), but also to handle php reserved words.
For example, without it you will never be able to use a URI like /products/list, since you can't create a list() method in your Controller.