The George Button that you have attached to Gertrude and Emma was actually married to Frances Crow. They married in about 1894 and had a daughter, Jessie, in 1895. They are all living together in the 1900 Census for Patton, Ford, Illinois.
Without solid proof of paternity for Gertrude or a marriage for Emma, it is better to leave the father as unknown rather than attach someone you think "might" fit the bill. I know it's frustrating not to be able to advance your tree, but I still feel your best option is to order death certificates and hope that there is additional information to aid your search. Then, when and if you can, a DNA test might prove very helpful.