A man isn't recorded as having given birth! In anything, you could argue that that would be more the case, if a legally male person was listed as mother on a birth certificate...
The child has a father. That is, a parental figure legally recognised as male.
(You may or may have a problem with people born as female sex who are later identified as legally male. And those people still being able to give birth. But it's what has happened!).
So that legally male parent is listed as father on the legal document recording: name at birth, date of birth, legal parents.
There is going to be nothing on the birth certificate to specify whom gave birth. Simply the absence of a mother. Presumably people whose mothers are unidentified also don't have a mother on a birth certificate. I'm sure this doesn't cause them confusion as to the fact they were born out of a uterus. Or change their relationship with their male parent if they have one.
Yes, people are trying to find problems with this situation that don't exist. Eg: assumptions about the way the child will be raised, what they will be told about their birth etc. Concerns re genetic lineology even though that isn't and has never been the purpose of the birth certificate.
There are some people who would insist it is less confusing for the person who gave birth to be listed as mother. On the other hand, you could argue that if your father (who raises you as a father, is legally recognised as male) is listed as "mother" on your birth certificate then that would be confusing! At the end of the day, it's not a typical straightforward situation. It is a little confusing. What a are you gonna do? Sterilise trans men? Stop people from legally changing gender? -insert mandatory reminder about intersex-
It's not a difficult situation to understand. The "weird" bit that is maybe hard to get around is that a legally male person can sometimes biologically give birth. Obsessing over the birth certificate won't change the weirdness. Weird doesn't have to be bad.
The way a child is raised is of paramount importance. Not how they were conceived, who gave birth to them, who is listed as parent blah blah blah. Why can't people care as much about real issues that actually affect children?