>  in my experience you are better off killing off everything and starting fresh.

Same here. Ironweed was my nemesis for years--used a weed wiper and Roundup on it many summers. Finally broadcast sprayed Grazon Next (?now Duracor?) and got 85+% control with just one spraying. Chemical and reseeding clovers was CHEAP compared to years of wiping with poor results plus poor grazing or hay in the heavily infested areas.