If it were me .... and as it happens I have the same thing here ... I'd do this - top the rushes continuously, tight, even in winter, in fact just before frosts is ideal, and mow low. Don't spray (you'll kill lots of beneficial micro organisms), don't plow (you will just end up unearthing many more rush seeds that you will need to let germinate and then have to kill off again).
In the spring harrow the hell out of it and sow an annual crop of deep rooted forage - crimson clover and vetch or something similar (as Farmerswife says, Cotswold seeds can help you here). Don't know much about hungarian rye grass but it might do what you want if its deep rooting, and an annual? - I'd still do a mix with something else though. Let that grow then graze it off, harrow heavily again and reseed with a deep rooting herbal ley. And where rushes persist, keep topping them. Select annual grasses/legumes in that first stage because then you can just graze them off rather than need to bring in machinery to til them back into the soil. Avoid heavy machinery.
Oh and bung on plenty of muck, as much as you can get. And lime if you can. Aerate regularly too (if you can).