I have understood that the reason at least in England for the clergy to name the putative father on a baptism was so the parish could look to that person to provide the financial support of the child, so it was public knowledge available to the local community.
The system was different in 19th Century NSW as there was no formal declaration of an "Established Church", and the administration provided funding for the Female Orphanage and the Male Orphanage from their initial foundations. (Again, these institutions were established during the 'reign' of Governor Lachlan Macquarie, a Scotsman, governor of NSW from 1810 to 1821. It was also during the Macquarie era that the practice of transmitting quarterly returns of baptisms, burials and weddings celebrated by clergy of all denominations were ordered to be forwarded to the NSW Chaplains. (I should note that NSW at that time included all of what is now the Eastern states of Australia, and New Zealand, and the South Seas Islands .... so a huge territory from a geographical perspective, and with Missionaries from various denominations sailing from London and etc).... so not just the settled areas in and around the penal settlements at Sydney, Port Macquarie, or at Norfolk Island, or in Van Diemen's Land.... So in the early C of E parish registers at St Philips, Sydney NSW, there are records of bdm events that were celebrated by various priests and clerics of many denominations at various locations in the Antipodes.
Re Baptismal records .... when the masses in say Ireland were suffering the 19th century famines and people were seeking to Emigrate, there were many emigration plans, including some organised to bring single females from Ireland to Australia. Those lasses needed ID, and the form used most often was their local parish baptismal certificate, newly issued by the incumbent priests at their local parish and the details included on their Passenger Lists which are now being uploaded to various commercial websites. Similarly, when families were migrating under other Assisted passenger schemes in the later quarters of the 19th Century, they too used their baptismal certificates for ID, perhaps it was a more reliable document at the time, or perhaps better recognised, or easier to seek out from the local Reverend....
I will look up and Add a link to a couple of the digitised Gazette entries showing some general orders.... your 21st century eyes may need to be put back into their sockets with some of the wordings....
HERE I ADDhttp://trove.nla.gov.au/ndp/del/article/627934 The Sydney Gazette 24 February 1810.
Whereas His Excellency the Governor has seen with great Regret, the Immorality and Vice so prevalent …… the scandalous and pernicious Custom so generally and shamelessly adopted of Persons of different Sexes Cohabiting and living together unsanctioned by the legal ties of Matrimony …………… and whereas such Practices are a Scandall to Religion, to Decency and to all Good Government; And whereas also frequent Applications have been made on the Part of Divers Women to the Court of Civil Jurisdiction ……..
So, you need to ask yourself
a) How did anyone in 1810 and earlier in Sydney Town, NSW, know that these Persons of different Sexes were living together unsanctioned by Matrimony unless someone was checking the parish registers
b) And so the civil administration were basically trying to avoid civil claims re Intestate estates and fatherless children seeking a benefit from such an estate
c) And what about those convicted persons who had married back in Britain…..
So, in respect of c) those marriages were effectively ended with the sentence of transportation beyond the seas but in respect of a) the parish registers were available for viewing, in a similar way to the present. You asked permission to view, and you waited until permission was granted, and you paid the fee asked for by the clergy.
There have been many threads about bigamy v re-marriage when transported beyond the seas but here is a starting point, so there were many reasons for accessing the parish records even at the time they were being made, and within the lifetime of the person about whom they were made....
http://www.rootschat.com/forum/index.php?topic=699561.msg5427556#msg5427556 Cheers JM