I'd be worried that pigs would do too much damage. They'll also fertilise that area massively - feeding the nettles and thistles that will recover faster than the grass. What's at the bottom of the slope? Any rosk of soil or muck getting in to watercourses?
I'm not sure how you could tell what state the seedbank is in, but if you wanted some clearance I might look at poultry rather than pigs, for a lighter touch.
You could also try feeding hay bales with deliberate wastage that gets trampled - the idea is that the seeds in the hay drop out, and the fibre of trampled and dunged-on hay traps some moisture and promotes germination, and protects the soil. It's a bit like using "green hay" to restore a meadow, but you're feeding your animals too. Doing this would also relieve some of the grazing pressure on the existing grass. If you look this up, it seems to be used more with horses currently.