I hope this isn't too late, but if the father's name was on the birth certificates, the index would show two entries for the birth; one for each surname, with the same page and volume number.
This doesn't appear to be the case with Eleanor Thorne's children. So there is no need to buy a birth certificate to find out if the father's name is on it.
If any of the children knew who their father was, they might have named him on their marriage certificate, when there is no restriction on an illegitimate child naming their father. But although this could have happened there is no guarantee that it did.