Our sheep and our ponies - the Fells in particular - seem to eat the rushes to a greater extent than do our cattle, including our Blue Greys (Galloway x Whitebred Shorthorn), Angus and Hereford cattle. However, the Blue Greys live on our 'fell' (rushy boggy area) with only a little hay over winter, so must be finding something to keep them alive.
They all particularly enjoy eating new growth.
So if you can top regularly, you'll get more grazing off.
As Dot says, the ponies probably eat rushes as much as they do because, being very good 'do-ers', they are always on very poor grazing. ; And we probably don't see much damage to rushes on the 'fell' as we've just a handful of cattle up there and it's a massive, and very rushy, area.
Irish Moiled look gorgeous - if there's a conservation reason to keep them, that's great, innit?