It's perfectly good English, and the meaning is broadly similar, but it would be changing both the vocabulary and the grammatical construction for no good reason that I can see. The literal translation works perfectly well here and I see no reason to change it.
Does he run every morning, should be acceptable and to say it's not, is surely a restriction of of choice
But that's rather "każdego ranka". When I see the Polish sentence, I'm not that sure that he runs every morning, I rather think that he never runs in the afternoon, in the evening or at night.