Olivers or maybe Oliphants? Latheron or Thurso

My apologies for a very long post, but I'm trying to sort out a line of Olivers (who may have been Oliphants). The time of confusion is the first half of the 19th century.

My 2nd great grandfather, David Oliver was born in Latheron between 1844 and 1848 and died in 1923 in Edinburgh. I haven't found a birth certificate for him, but according to his marriage certificate to Esther Henderson (30 Dec 1864, Thurso), his father was George Oliver, police officer, and his mother was Elizabeth Oliver, maiden surname Sutherland. David's 1923 death certificate has his parents listed as George Oliver (Shepherd) and Betsy Oliver, afterwards Hamilton nee Sutherland. But...family lore is that George and Elizabeth were never married, that George may have been a young shepherd from the south.

After giving birth to David about 1848, Elizabeth married a James Hamilton in 1849, and had eight more children, living in Bower until moving to Edinburgh by 1881. The 1851 census has her son, 4 year old David Oliver, living with his Sutherland grandparents (George and Margaret (Sandison)) in Latheron.

There is a George Oliver who fits the general profile--he was born in the south about 1833, and was living in Thurso by 1841. There's no direct connection I've made between this George Oliver and Elizabeth Sutherland. He married a Johan McKenzie in Thurso in 1853, five to seven years after my David Oliver was born, and four years after Elizabeth married James Hamilton. George and Johan moved to Australia, to New South Wales with their two children in 1856. I've found plenty of information about George in Australia, and he could be my third great grandfather, but...

I've had quite a few DNA matches who are descendants of an Essie Oliphant, born in Adelaide, South Australia in 1879, daughter of a George Oliphant, born in Wick in 1848, son of a William Oliphant, also from Wick, born 1821. I can't find a George Oliphant the right age, and I can't figure out any connection beyond the DNA, but it does make me wonder.

The key pieces I'm trying to answer are:

Who was David Oliver's father? If it wasn't George Oliver born 1833, was it an Oliphant? Which Oliphant? (I've been making an Oliphant tree, but just can't connect it to my people). And why does David's 1864 birth certificate list his mother as Elizabeth Oliver when she'd been married to James Hamilton for over 10 years by then?

Many thanks for any advice or information!