The effect of string stretching is pretty easy to verify. After putting on a new set, tune them to pitch and the stretch each one side to side and away from the fretboard a few inches. They will all be at least a quarter step flat afterwards, usually more. Tune them to pitch again and restretch and they will be less flat.
I'm not sure stretching matters for the long term tuning stability of the strings but definitely has an effect on the immediate tuning stability. Players who don't bend much and fret/strum lightly might not need the added stability but I like it.
Also, I've only ever heard the old strings sound better thing about flatwound strings, particularly on bass guitars, in which case it's true as long as "good sound" to you means a serious lack of treble.
"Well, call me a rebel, but I don't stretch my strings and I will go one stage 30 minutes after putting on new strings"
Which would probably work.
Back in the day I would break strings on stage. Yes, stretching strings helps a lot in that case.
Of course you can only do so much of it before the rest of the band has milked the specially rehearsed for just this occasion "intermission music”, but it really does help.
Another odd one is over tune new strings to get stretch, which often gets turned to don't put new strings on a guitar the same day as a show.
Well, call me a rebel, but I don't stretch my strings and I will go one stage 30 minutes after putting on new strings.
These are fine rules of thumb, I suppose, but as you are saying, a lot of nuance is involved.
The only thing I'll flat out disagree on is that old strings sound better... No just no.