Depending on what your weather conditions have been this year, spraying may or may not be affective. Aroiund here, now the pigweed responds pretty well, lambsquarter and ragweed really respond quickly. We're now into our second growth of thistles and they respnd almost within a day. If the weeds are just atarting to pollenize spraying will work fairly well. If they've set seed it's a waste of money to spray. If they haven't reached the pollen stage they'll respond best.
If you aren't soncerned wit6h a grazing ability I would use a less succulent type of grass. Juicey grass will stain and it can be a real joy trying to get it off. It will also give them the squirts and those tail swish stains are a blast too.
That said, maybe bluegrass or one of the lawn types of fescue. Low growing and will sod over to cut down on some of the mud/dirt situation.
Rather then plow I would disk the snot out of it, drag it to smooth it some then broadcast thr seed and roll it in.
If you're going to disk anyway, don't bother spraying. If you spray with something that will kill the roots, you have to wait a while for it to transmigrate to the roots, a week or two, sometimes a month.
If the weeds aren't real tall you can spray with roundup and seed a couple of days later, but you would probably be better off using a no-till drill in that situation.