I really don't think people are missing the point. This is Legal not AIBU and the moral ins and outs about your child's parentage /potential contact is fairly irrelevant. There are just some basic procedural facts that often get conflagrated when they are very separate issues.
Being named on a birth certificate gives parental responsibility for the child. Equal responsibility with anyone else named on the certificate. (Unless PR has been removed by the courts)
Not being named when PR is wanted requires an application to the court for leave to apply. The court will ask on what basis. If it's parentage then you the mother will be asked if this is correct. If you deny parentage the court will order a DNA test. If you refuse- you will be in contempt of court . Not a place anyone needs to be ! Especially if you want the court to look upon you favourably.
Once the DNA confirms what you already know - he can apply for PR. It is extremely rare for a biological father to be refused. It is simply a fact. If he is a father then he should have parental responsibility. (Unless less proved otherwise by a court) .
Once he has PR an application for some form of contact is almost guaranteed. This can run from indirect (letters, phone calls ) , through supervised visits all the way to shared care and beyond. This is entirely dependent on the evidence produced by both parties, 3rd party reports (schools sw etc) and a judge's decision of what is in the child's (not the adults) best interests.
Maintenance is an entirely separate matter. Your child has a right to financial support from both parents. Non resident parents have an obligation to maintain their children to a minimum level set by government, taking into account their individual circumstances.
Child maintenance does not equal access. He can be assessed at the highest rate of maintenance CMS and be refused contact entirely by a court despite having jumped through the hoop to get PR.
Not naming a father on a BC is at best a stalling tactic for a determined man. It's fairly easy to navigate the hoops to get it and can be done himself costs £232 and can be reduced for a low income . However for a feckless, game player who isn't really interested in being a parent .. it's a convenient obstacle. They probably won't be arsed to climb over..
He is either going to go for PR or not. You might as well get your child the financial support they are due.