When I started mowing walking paths in the pasture a lot of the woody plant "stumps" kind of went away and weeds were replaced with soft grass and clover. Of
course the
cattle like best to graze the mowed paths so they may have trampled them. So maybe this isn't helpful information. LOL!
When I first got pigs they dug a lot and made the ground so lumpy it's still hard to mow there (with my push mower) because the wheels are always getting stuck in ruts. It's just a paddock so it doesn't matter that much, but for a lawn you'd want it less lumpy, I think. If you think you'd like some sheep I think they'd like to eat most of the weeds for you and what I hear is they prefer them and will leave a nice grassy sward when they are done - they don't make ruts and their poos just kind of disappear (pig poos are similar to human ones).
Don't get me wrong, I love pigs and love having them and think everyone should have a couple, but for a lawn, I don't think they're the best animal.
I think it will go faster if you have some way of mowing or brush hogging from time to time - even animals who eat a lot of brush prefer to eat the new soft shoots over old woody growth, and in spring it grows so fast they can't keep up unless you have a pretty high stocking rate.