I do think Neale1961 has the right idea for you to obtain the death certificate of the John Davis who died in 1871 in Beechworth. It really doesn't cost a lot and will - hopefully - confirm or negate.
But meanwhile just to confuse you more

I have a few links and thoughts for you.
Following on from Neale1961's findagrave memorial link, the son of John Robert Davis who is buried with him, Denis, was born at 'CLAR' and this stands for any of the following according to Victorian BDM abbreviations list: Clarendon, Claretown, Clarinda and Clarkefield. See Vic BDM Reg No 5799/1876 These CLAR place names can be all found in Victoria, largely near Melbourne or North of Geelong. But of course it might also stand for County CLARE. (Not matching the county Down as you found on FamilySearch). It might stand for another O/S place, I haven't searched.
i.e. If Denis was 20 when he died in 1876, it likely means this John Robert Davis was in or around one of the CLAR places in 1856.
This is a Trove link regarding the Inquest of John Robert Davis's death. The article says a John Davis had been admitted to the Lunatic Asylum (in Beechworth presumably) from the Ovens District Hospital. The Ovens district is virtually the same area as Beechworth/Tarrawingee/Bright/Harrietville - all the places so far linked to this particular John Robert Davis.
https://trove.nla.gov.au/newspaper/article/196417574Here is a link to a brief history and description of Beechworth. You will see that the Chinese immigrants figured greatly in its early history.
http://melbourneblogger.blogspot.com/2009/04/beechworths-victorian-architecture.htmlWas it the same John Davies (John Davies in the passenger list) who arrived in Australia early 1869? Just to confuse things there is also mention of a John Davis from Tarrawingee in an 1868 Trove article, before the other Davis arrived in 1869, who had to forfeit his licence (it doesn't specify what type, might be mining):
https://trove.nla.gov.au/newspaper/article/197437081And after John Robert Davis who died in February 1871, there is mention of another John Davis having an accident at Growler's Creek (in the same area) in April of 1871. So many John Davises - and in a small population! Of course, the second John here might be John Robert's son...
Following on from the mention of Tiddledee-Addeldee, here is a link to several Tiddledee-Addeldee mine articles, where a John Davis is the mine manager.
https://trove.nla.gov.au/search/category/newspapers?keyword=Davis%20%2B%22Tiddledee-Addledee%22&l-state=VictoriaIf it is your John Robert Davis who died in 1871, there is information of sorts to be had - so again, I think it's best if you confirm yay or nay. Good luck