Could I please have any source material you might have to ensure that the correct children are with the correct parents please, as this is where my roadblock started.
All primary source material is at
www.scotlandspeople.gov.uk.
1841 census, in Girvan: Alex, 30; Mary, 30; Jane, 10; John, 8; Barbara, 6; Agnes, 4; Alex, 2; Sara, 0.
1851, Kirkoswald: Alex, 43; Mary, 43; Barbara, 17; Agnes, 15; Alex, 12; Sarah, 10; Margaret, 8: Mary, 5; William, 3; Elizabeth, 1; Ann Dick, 0.
Using the facility to search for deaths by mother's maiden surname, there are possible deaths for all except Barbara and Sarah, though I am dubious about James and Alexander, as both of those deaths are in Perthshire, not in or close to Ayrshire
Jane may have died in Ayr in 1905 or Maybole in 1909, but there are two other Jane Thomsons with mother's maiden name Clark of roughly the right age
John may or may not be the one who died in Blairgowrie in 1870
Agnes died in Barr in 1912
Alexander might or might not be the one who died in Logierait in 1905
Margaret is probably the one who died in Girvan in 1929 but there are four other Margaret Thomsons with mmn Clark of roughly the right age.
Mary died in Sorn in 1877
William died either in Maybole in 1922 or Dumfries in 1923
Elizabeth died in Hurlford in 1905
Ann Dick died in Bute in 1931
You will be very wise to follow the advice already given to ignore all online trees, try to forget anything you have seen there, and start from scratch with the primary records.
Though as you are descended from the Largs family you can probably safely dismiss all of the Kirkoswald/Girvan family as unrelated.