Not true at all. You need to read more on the Church Fathers. BOB was explained as a sacrament; BOD has never been.
I cited that 5th century theology manual that for a long time had been attributed to St. Augustine, stating quite clearly that BoB was the only exception to the normal Baptismal ceremony because all of the Sacramental elements (aka matter and form) were present. St. Cyprian called BoB a SACRAMENT and explained that the angels pronounced the words of the form over the dying martyr, whose blood served as the matter for the Sacrament.
We have several Church Fathers rule out BoD by saying that martyrdom is the ONLY EXCEPTION to normal Baptism. But even then they held it was no real exception because all the elements were there.
This pretending that the BoB Fathers accepted BoD by inference is absurd and dishonest. Even St. Alphonsus admits that they're different, with BoB acting "quasi ex opere operato".