Honestly, the amount of people who assume that unmarried mothers can just turn up at the registry office and go "yep, Baby Mother's Last Name, born in X hospital on Y date... yep, that's right? No, no, the dad and I aren't married. No, he's not here - he's busy/working/ill/done a flit/was a one night stand... His name, yep that's Fred Bloggs. Yeah, just put him down...". When actually? Joe Bloggs is the biological other parent.
We can't.
It's very often not that we don't want our children to have full birth certificates, more so that we need the father actually physically sat next to us at the time for this to happen. Legally, if the parents are married, either one can register the baby. In the UK, if the parents aren't married, then, legally, only the mother can do so.
This is why, when a bloke's name is added to the birth certificate at a later date, courts are involved, along with paternity testing, because, again, the onus is on the father to prove that he actually is. It is to protect men and the children alike.
My son is 19 now. Doesn't have his father on his birth certificate. Not because I didn't want him there - because he didn't want to pay the court's fees in order to have himself added. We're (obviously) not married at the time DS was born. At 3 days old, the midwives flagged up an urgent medical issue, our GP stubbornly stuck to the rules of not being able to check it out without a birth certificate (I forget why now), I had to get an urgent/emergency appointment to register DS' birth... and his dad? Was not picking up his phone (we didn't live together, were nearly 30 by this point, and I had a anxious 8 year old to deal with, too. Whilst 3 days post-partum.). The registrar (who was lovely) said, when I double checked, that yes; you and dad just fill this form in, the judge will sign off on it, DS gets a new birth certificate with no blank spaces...
Told his dad this. Repeatedly. Dad looked into it and discovered that he would be responsible for the courts costs. He said he'd think about it. Every time I asked after that, he'd change the subject, until we split when DS was almost 5 (he'd been cheating)... and his father tried to say that I had deliberately not put his name down when I registered DS' birth. Ex-FIL got a sharp shock that day, because I told him the truth - and ex couldn't deny any of it, when his father checked... and discovered that I was right.
Having his father's space blank on his birth certificate hasn't impacted DS' life in any way (neither has it my older child, but her biological 'father' was abusive and keeping him off it, 28 years ago, was done to protect her from abduction to the Middle East by him). Yes, it's frustrating when people think our system works like it seems to in America, where apparently a mother, married or not, seems to be able to have a baby with Joe Bloggs, but name his wealthier brother, Fred Bloggs, and no one checks.
@wooldryxptto - as long as your DD knows who her father is, and he maintains his half of her... and they build a strong relationship, that really is all that matters. It isn't like schools or employers are going to shun her, or people sneer at her like they would have 40 or 50 years ago. I suspect that his reluctance (because it said on the form 20 years ago, and common sense dictates, that a court needs a hard copy and both parents present... otherwise you could have filled it in with some random bloke who isn't biologically anything to do with your DD. Not saying that you would, but some do) is more to do with whatever caused him to abandon you when you were pregnant and needed his support.
and lots of good wishes.