Alright I take your point about the cowboy being gunned down rather than dying of the pox, but all the derivatives of the Unfortunate Rake refer to someone dying an untimely death.

If you look at the lyrics of St James Infirmary in the DT, there is reference to bars, gambling and chorus girls. So although it is not explicit, it is implicit from the context of the song that the dead girl has died an unnatural death so syphilis or violence are not unreasonable assumptions.