Annie's parents were Edmond and Mary according to the online tree, so that throws things off a bit in terms of naming pattern, there being no son Edmond, and Mary being the third daughter, rather than the first. The nickname Fanny is more English and Irish than it is Scottish. The name Alfred is also very rare in Scotland at that time. They may be just a reflection of the fact that Annie was Irish, or... John's family may have been living in England or Ireland and he may have been born there.
These are the children given in the tree:
Elizabeth "Isabel" 1867 Santa Catarina, Brasil
Frances "Fanny" 1868 San Javier, Prov. Santa Fé
William (no birth information, do you have any?)
Alfred 1874 Rosario, Prov. Santa Fé
Mary "Polly" 1876 Rosario, Prov Santa Fé
I found Alfred in the 1895 Argentina Census on FamilySearch:
Alfredo Thompson, age 20 is single and living in Rosario
I am well aware of the problem trying to find John's parents and birthplace, having spent years trying and failing to take my Stewart ancestors back to Scotland. Until you have exhausted all possible resources in North and South America, looking for his birth in Scotland is like trying to find a particular needle in a large haystack! And you won't know for sure it is him even if you do find him! (Unless you can find a newspaper account of him being in Argentina-- now there's a good idea!)
You may have to also consider that he may not have been actually born in Scotland. It is fairly common to find family stories about origin that collapse several generations. One of my great great great grandmothers was referred to in the county history as a "Scottish lass." So far I have found that she, her parents and her grandparents were all born in America… so the Scottish connection keeps going farther and farther back!
Reminds me of my mother-in-law who used to speak of the "English" people in certain parts of France, whose ancestors came from England during the 100 years war (which took place from 1337 to 1453)! But they are still "English" to her...
All the best,
Sarah