Is it possible that there was another Elizabeth South in the parish who gave birth to the children who have no named father? From what you say Daniel could not be the father and it is quite possible that Elizabeth strayed while he was away, but why did the Parish Register say that she was a spinster, when it is quite clear that she was a married woman. I presume that all these entries were in the parish of Birch.
From experience I have often found two women of similar ages with the same name in a village, although probably related. I think that you would need further evidence that all these children belonged to the one Elizabeth before you could say that they were all siblings. There could of course also be two Daniel Souths.
If a couple had two sons it would be normal for both of them to name a son after their father and a daughter after the mother. If they continued to live in the same village then cofusions would arise.
I notice that in 1841 an Elizabeth was living alone with daughter Elizabeth 10 and son Daniel 5. By 1851 it appears that Elizabeth was the wife of a William Faskin? and Daniel South 15 is called a 'son'. Even if Daniel senior is his biological father I presume that he was deceased before then and probably before 1841.