You may be right, however, reading many of the well informed posts on this thread in relation to the welfare of the deer, it appears from here, that there are good arguments based on the welfare of the deer, without reference to individual shooters mental well being.
The argument has been raised, and whilst your faith in 'the government' is laudable, PETA (and similar organisations) for example, are likely to seize on any arguments they can to reduce shooting in all its forms, by whatever means they can.