The OP does have legal rights to the child. He has as many rights and responsibilities as the mother simply by being on the birth certificate; he does not have to go to court to get them.
That article doesn’t say we have notaries, exactly, but that we have ‘notaries’: people who have another job title (solicitor or commissioner for oaths) who perform some of the functions of a notary in the US, but not all, and not all solicitors will perform them.
Since they certainly appeared to be in control I really, really doubt that. They got a lot of press because they were the ones who were talking; the ones who published books, who got laws passed, who held conventions. If the feminists who disagreed with them were the silent majority, then they were very, very silent.
I think, though, that we can agree that UK notaries (or solicitors or commissioners, as the case may be) will not perform the same functions as Maastricht is attributing to a notary as she knows and experienced. Which likely closes off that line of recourse.
I will say that as a Canadian lawyer (not only a notary but also a barrister and solicitor, for those who are keeping score), I have seen similar agreements drafted and adhered to by the parties, work. If the parties agree that the best interests of the child are served appropriately by the agreement, and they are, and the parties adhere to the agreement, then there is generally no court intervention necessary. In my experience, it is when a party wishes to vary the agreement (most often, contrary to the best interests of the child) that court intervention might be necessary.
For now, OP, it might be helpful to have your agreement with your ex properly drawn up, and attested to. A lawyer can help you with this, and advise in such a way as to ensure that your agreement does not run afoul of the provisions of any existing statutes.