Yes molehills are useful in the garden, but there are a couple of problems with having a lot on grazing pasture. One is that they can bring listeria to the surface, so as sheep graze they will pick up some soil, or soily grass and run the risk of listeriosis. The second is that a ewe can drop her lamb onto a wet molehill, so the soil sticks to its wet newborn wool and is very difficult to get off. Very heavily pregnant ewes can 'cowp' (or whatever you call getting stuck upside down in your area) if she gets it wrong or is already on a slope. Lambs love molehills though as they are big enough for them to shelter from wind and rain.
A tale of a molecatcher: when I was growing up in 1950s Norfolk, there was an old guy who was to be seen every day on the rattliest old bicycle you can imagine, dressed in a filthy long coat and with a couple of saggy bags hanging off the handlebars. Turned out he was the molecatcher . I think he lived in the woods, not a house. He came to ours sometimes, not for moles as we didn't have any, but just for a gossip. You could smell him long before he arrived and he was rather a dour old chap. My mum asked him about the one very long finger nail he sported, which turned out to be for defence!! He saw himself as scraping out someone's eye with it if they attacked him. Very sadly he disappeared, and we found he had been crushed to death against a wall by a lorry as he cycled along - his defensive finger nail was no use at all. Looking back, I think the coat was probably made of moleskins, the bags too, and the smell a mix of mole and well, man.
Oh I nearly forgot. We have loads of moles here and one day the gamekeeper from the big estate stopped and offered to catch them. He charged a pre-agreed £60 to clear the lot, which required several visits and bagged about 20 moles, so at £3 a mole it looks like that was a bargain. Of course he didn't catch quite all of them.........