Someone suggested up thread that 'there is only one set of church records for Falkirk'. This opens up a whole can of worms

The extant registers of the Church of Scotland were collected centrally in New Register House in 1855 when statutory registration of births, marriages and deaths began. Later baptismal registers were never collected centrally and if they have survived are likely still to be with individual churches. In particular they are not indexed in either Scotland's People or in the International Genealogical Index.
With a surname like McGill, and an Irish connection, there is a possibility that the family were Roman Catholic. The RC records after 1855 are indexed on SP.
I see that Maggie McGill, in the 1881 census in Troqueer, is aged 8, which is exactly the age that Mary, born 16 November 1872, would have been on the date of the 1881 census, and I wonder whether 'Maggie' is a misrecording of 'Mary'? That death record that Monica found may help to answer that, of course.
Did Mary/Minniepersonally write that she was born in Falkirk? If it was her husband, could he have got it wrong?
Looking on the bright side. I think there might be a slim chance that the Falkirk church records haven't been digitalised yet.
I think you can be 100% certain that the post-1855 Falkirk baptism registers have not been digitised yet.
When the Mormans went round churches in the 1980's to film their record books not all parish churches gave their permission.
The Mormons did not go round churches in Scotland. They microfilmed and indexed the pre-1855 registers of the Church of Scotland which had (as described above) been gathered in New Register House, plus the statutory birth and marriage registers from 1855 to 1874/5. This was in effect a 100-year cut-off being operated by GROS in the late 1970s.
I believe that it is true that the RC church refused to allow the Mormons to film and index their records, and I am also under the impression that in England, where many churches still retain all their own registers, some churches also refused. However this was not the case with the pre-1855 Church of Scotland registers.
The flaws in the Mormons' index are to some extent perpetuated in the SP index. There are, for example, sundry omissions from the IGI, and most of there are also missing from the SP index, though SP will eventually correct any omissions to which their attention is drawn. This isn't going to help you find Mary/Minnie McGill, because it applies solely to the pre-1855 C of S registers.
I remember when I started fam.history in 2002 that a few genealogists had already set up scotsorigins (part of britishorigins) and had organised a contract to offer digital images to amateur family historians. Not all church records had been filmed when scotsorigins lost the contract and it was given to scotlandspeople.gov.uk
The contact was not taken away from Scots Origins and given to Scotland's People. Scotland's People is a government agency, and in the early days it was SP who subcontracted Scots Origins to run the online service. Later SP took the service back in house, using Brightsolid for the technical aspects of the service.
I've had a look at the Falkirk Town Council's family history page and although it mentions all pre 1855 church record/registers have been transferred to the GRO central archives it doesn't actually say that all later ones have been transferred and it may be worth sending them an email asking if this is the reason you can't find the baptism of your ancestor
If they say all books have been transferred - it may be that the GRO haven't filmed the register yet.
See above. The post-1855 C of S registers have not been transferred. Some may be in the care of local archives but most are still with the churches.
The GRO is the agency which deals with BMD registration in England and Wales. The corresponding agency in Scotland is the GROS. As far as I know, and I'd be happy to be told I am wrong, there are no plans at present to centralise/film/index/digitise the post 1855 C of S registers. There are, however, plans to index the assorted Free Church registers which are held in the National Archives of Scotland, not by the GROS.