I've not read full thread (so sorry if this has been repeated). Even if a father is on birth cert, it won't necessarily mean that CM is being paid or that the father is a decent and hands on parent (can be quite the opposite). It does mean that the father can have a say (legally, as he has PR due to being on the BC) on things like where the child lives, where they go to school, contact, medical treatment, travel/school trips etc. If the father is a decent guy, then great, there's no problem. If he is not, then this can cause all sorts of hell. I've seen too many threads about where an abusive ex has had the children for their contact days and refuses to return them. Police can't really do anything as the father has PR. It sounds like hell. Also when fathers have been abusive and yet contact is still expected to happen (I've seen that with some of my friends...and to an extent, myself).
Example: With my XH, we were married so I was able to add him to BC without an issue. When we split, he'd turned into a monster (abusive....towards me). It wasn't safe at that point for him to have unsupervised contact with DC2: not because he would deliberately hurt DC2 but because (at that point) he was drunk/high most of the time, would drink/drug drive and his behaviour was unpredictable and (therefore) unsafe. It took a Prohibited Steps Order to put some things in place for supervised contact. Had he not been on the BC, it would have made my life back then a lot easier (and I wouldn't have had to shell out thousands for a PSO). If I had not had the PSO, he would have been within his rights to just pick up DC2 from school without my knowledge or consent, and the school would have been powerless to stop him (because he has PR). To be clear, I was never looking to completely stopping contact. I was only stopping contact if he was drunk/high and I was asking for supervised contact during that time. He's fine now, and that's great. But back then, it was hell. Luckily the Judge saw this, agreed with me and granted the PSO. He also ordered a CAO (Child Arrangements Order) be put in at the same time (which said that DC was to reside with me full time, amongst other things).
FWIW, my eldest DC's dad walked when I was pregnant (and we were in an LTR of some years). He refused to have anything to do with DC1 for about 3yrs. He wasn't on the BC as we weren't married and he wouldn't show up for the birth or registering the birth. He eventually saw the light, got back in contact with me and we sorted things out (didn't get back together but I made it clear that if he wanted any sort of contact/involvement, he was to start acting decently towards DC1 and being polite/respectful towards me etc). It was only after some years of him proving himself as a decent father (and to be fair to him, he did) etc, that I sought legal advice and had him added to BC.