Andrew Burton is a good starter that currently has the problem of being outclassed by other dudes with similar abilities. Travis Moone gives the same bullet rating for cheaper, and a full mulligan is better than a single rummage. Genesee "Gina" Tailfeathers actually gives you card advantage. Philip Swinford can trigger more often and has an influence. Rev. Perry Inbody has a better skill rating and an ability that is relevant on every turn. Also all of those starters are at a higher value so may not be as punishing should they cycle through your deck.
The ability that I am really looking towards however is the ability to add a bounty to a dude before the game begins can create some really explosive starts for Law Dogs.