The soap character I particularly associate it with is Irene in Home and Away. She definitely belongs to a particular demographic (older woman, a bit rough around the edges) and she isn’t native to the show’s setting so they may be deliberately using it as a marker for the character. I don’t know where she’s supposed to be from but.
It’s definitely not a socio-economic linguistic quirk in SA, WA or any parts of Vic that I’m familiar with. I stand corrected in that it’s in use in QLD as well, but it’s certainly not a nationwide linguistic quirk at all.
Stand up comics use it as a a way to imply that he is “just saying something” presumably funny or witty, and he’s got more to say but the laughter prevents him from continuing, and he graciously drops the topic. He then goes on to another set-up.
Once you notice it, you realize how ingrained it is in the comic’s craft.
This is in America, at least.
I think it might be similar to the use of “only” in Indian English. It has no grammatical role, it’s difficult to define it’s meaning, and it’s mainly used as an intensifier.
“What to do? He is like that, only, no?” (“Nothing can be done about it. That’s just the way he is. Am I right?”)
Hingmy.
“But” at the end of a sentence is definitely a Glasgow and surrounding areas thing. Nobody says it in Aberdeen. “Like” at the end of a sentence, serving no real purpose, is pretty common though.
No, as other have said, the OP usage in Scots and Australian english does not involve an ellipsis and it is not merely an intensifier or excess verbiage.
It is not “I don’t mind looking at TV but…” I don’t doubt that usage exists. That is not what the OP is talking about, but.
The usage the OP is describing simply involves using “but” instead of “though” at the end of a sentence. It is a purposive grammatical construction, unconventional as may be.
If you haven’t heard this then I expect it’s hard to fathom, but it is certainly not merely people trailling off their sentence.
For the ending ‘but’, sounds like it is used like it should be at the front of the sentence.
Interesting, and now I’ll know what someone means if I hear them say it.
You know now that you mention it, I recall an old Aussie sitcom “Hey Dad” where the secretary, Betty always did this. Betty was from the country and the other characters would make fun of her for dong it.
I’ll see if I can locate some examples on YouTube.
It’s definitively not a trailing off thing though
Interesting town. I wouldn’t want to live there but.
Says the Novocastrian 
OK this is a perfect example of how I hear it being used
Betty’s a nice girl. A bit dim but.
When I was a child in Corby (England), in the 1950’s, about half of the people who lived there had actually come from Scotland. I used to start arguments with my friend, who was from Edinburgh, when we were about eight years old, because she ended sentences with ‘but’ (meaning ‘though’). It definitely was not a trailing off kind of ‘but’! However, having moved to Corby when I was six, after a few more years of living amongst all these Scots, I got into the habit myself, and all these years later, I sometimes hear myself ending sentences with ‘but’. My husband happens to be Scottish, from Montrose. However he never ends his sentences with ‘but’! He does say some other things which sound a bit strange, but!
If I heard anyone doing this, I would kick them in the but.