I see this is your first posting on Rootsweb, at least under this user name. Your posting was sufficient for this research (most just put in a name with no other information, making common names such as this one difficult to identify). You could have also identified the name of the sister and her death info, if known. That would help to identify if she was or was not alive at the time of his death. If she had already died, the likelihood of his having an obit would be diminished...unless she had family (children/grandchildren) in the area.
There is a listing for him in the Idaho Death Index with the data you provided.
The Daily Nevada State Journal (Reno, NV), April 22, 1893.
Nine miners were cut off from escape by a fire in a shaft of the Butte & Boston Company at Butte, Montana, and all hope of rescuing them is abandoned. The doomed men are Edward Pasco, Richard Andrews, Sam Ravetto, Frank Geerard, RICHARD TREMBATH, Thomas Gray, James Fetto, Antonio Beara and F. Puglis.
--I did a search for some of the others (less common named) and found none of them in the 1900 Census. So, it "appears" none survived. If this is the correct Richard Trembath, he would have had to have survived alone OR the others that I checked died before the 1900 Census.
The 1900 Census for Butte City, Silver Bow Township, MT
Richard Trembath, born Sept 1857 in England, immigrated in 1872, status "Na"-Naturalized, Ore Sorter.
James, son, Sept 1877 in MT
Annie, dau, Dec 1882 in MT
--If this is the correct person, his DOB is about nine years different than the person in the Death Index. Mistakes in both indices are often wrong.
--This is "possibly" the one identified in the news article? Also see below in the family tree information.
The new York Passenger Lists show a Richard Trembath, born about 1848, arriving 21 August 1910 from the port of Southampton, England, on the ship "Philadelphia." He was from Butte City, Montana, age 62. Appears to have been traveling with John Henry Hicks, age 36. Richard lived at 537 N. Franklin, Butte City.
The 1910 Census for Butte Ward 2, Silver Bow
D.F (Dennis Francis ?) Egan, head, born in Iowa to both parents born in England, married 16 years,
Kate, wife, 35?, born in Utah to both parents born in England, mother of 2/2 living
Sylvia, 15 (Married Edward Michael Schaad)
Richard Trembath, 63(1846/1847), widower, born in England, immigration "Un"(Unknown, so someone other than he provided the info to the enumerator).
The Butte City Directory, 1902, identified Richard Trembath, miner, living at 537 North Franklin.
Family Trees posted online, show a Richard Trembath, born about 1860 in St. Just-in-Penwith, Cornwall, England, died in Butte, Silver Bow County, Montana in 1901 in a mine accident. His father was Hugh Trembath, 1824-1911, mother Amelia Jane Walls, 1825-1908. He had a daughter Annie...this matches the information on the person found in the 1900 census in Butte, apparently another wrong person.
--Be VERY CAREFUL in using posted family trees for information. MANY do not have identified source information. MANY have wrong information. This particular one had four siblings of Richard named Amelia Jane with varied births and deaths.
I don't see anything in your posting that says "his deceased sister lived there." She was "possibly" alive when he went to visit. The other person appears to be reading something into your post that isn't there. It "appears" he was living with his daughter in 1920 and then "might" have moved to live with his sister; or, was in the act of visiting. The publishing of an obituary depends on several things. First there has to be a newspaper publishing in the local area; sometimes obits were/are published in a distant paper IF it is usually read in the area. There has to be enough family/friends in the area for the obit to be read by those knowing the person. The person had to have lived in the area long enough for persons other than his family to know them. The family would have to pay for the obit. Back in 1919, obits were not as popular as today, nor as elaborate. An obit then was more like a death notice today, just the name, date of death, place of funeral; more of an advertisement for the funeral home than an obit. A "true" obit was usually written by a newspaper reporter about the famous, infamous, rich, or those deaths caused by crime or accident (fire, automobile).
I suspect you would find an obit for this person in Montana; but, don't be surprised if one is not found in Idaho.