Hi there,
Sorry for long post
Further to the Registers at St Phillips i) General Orders issued by Gov Lachlan Macquarie provide instructions regarding the civil administrations requirements for the recording of baptisms and burials:
http://trove.nla.gov.au/ndp/del/article/628072 Sydney Gazette 22 Sept 1810
ii) Legislation (Georgii IV No 21) Gov Thomas Brisbane regarding the recording of and transmitting of records of Baptisms, Marriages and Burials, solemnised in the Colony of NSW or any of its Dependencies and towards the end of this cutting :
“(in Case the same shall not have been solemnized or performed by the Established Minister of the Parish wherein the same shall have been solemnized or performed) be transmitted, within One Week from and after such Solemization, to such Established Minister; and such Certificate shall be signed and entered by such Minister, in like Manner as if the same had been by him so solemnized or performed” .....
http://trove.nla.gov.au/ndp/del/article/2184654 Sydney Gazette 7 Nov 1825

From several paragraphs in the Introduction to the book “The Catholics of Nsw 1788-1820 and their Families (JH Donohoe) :
ii) ...
“It is most misleading for a researcher to believe that the Church of England records exclude Catholics. It does not.... It should be remembered that those registers were never exclusive to the Church of England. .....To suggest that any person found recorded in them was of that faith only is a misnomer”..... 
From info about the Editor of above book
“.... a career Public Servant who has spent thirty years in various areas of Public Administration including .... the NSW Registry of Births, Deaths and Marriages”. So the register at St Phillips, contains far more than just records of C of E BDM celebrated at St Phillips, but as an aside
“Persons who may wilfully insert false or forged entries, to be guilty of felony, punishable by transportation for fourteen years.” http://trove.nla.gov.au/ndp/del/article/2446837 Colonial Times (Hobart) 23 Dec 1825

Please don't ask what happened to the clergy who failed to transmit TRUE entries, it probably depended on the proving of wilfulness, v civil law v ecclesiastical law but at the moment I haven't got ready access to my notes as I have lent them to an elderly family member for their comment. Relying on memory, I doubt there was any contemporary checking of original records against those in St Phillips registers, and I am aware of the hesitance of some clergy of various denominations providing the Established Church with any information about their own congregations. Compound that with the then prevailing conditions re mail, civil regulations, lack of funding from civil authorities for clergy to perform civil admin functions etc ... I think it would have been a dilemma, but without those St Phillips registers,
Cheers, JM