Birth cert is a record of fact.
This person birthed that person.
The only people who can birth are women.
The name for women who have given birth is 'mother'. It is not 'father'.
Freddie can be a mother who choses to see Freddie's self as a father, and Freddie can compel Freddie's child to see Freddie as their father, and Freddie can present Freddie's self as their child's father - but Freddie is the child's mother, and should be recorded as such on the CHILD's birth cert - that info belongs to the child, not the parent, and the child deserves accurate info on their birth cert.
Whether or not 'mothering' is surrounded by sexist assumptions is a whole other question, and irrelevant to the issue of a child's documentation.