The best quality mince on the market is steak mince. I'm not sure of the best cut to use for mincing but I would be leery of a roasting joint - much better to roast it. I would probably use braising joints my self.
Or rather than get supermarket mince with nasty stuff in it, go to a butcher or meat farm shop and get your mince there. My local farn shop sells cheap mince made from the offcuts of butchery (£11.50 for 10lb) which is fabulaous, fairly low fat and tastes lovely - I would even be prepared to use it for a tartare dish.
"For to be free is not merely to cast off one's chains, but to live in a way that respects and enhances the freedom of others." - Nelson Mandela