Define the "natural range" of a trumpet.
It's impossible to define because it's different for every person. what feels natural to one may seem ridiculous to another. Wayne Bergeron is a perfect example. He sounds completely natural up in the stratosphere. Not strained, or harsh.
There is nothing wrong with extending your skills to the extreme as long as other areas of skill don't diminish to achieve those results.
I was always taught it was best to be well rounded in your skill set, instead of a one trick pony. If you can hit a double high C with good tone...That's awesome....just be sure you aren't sacrificing tone or technique to get there.
I honestly think the high note thing is so common (particularly among young players) because it is a well defined goal. Hit "the" note. Once hit...it typically is celebrated by young peers as an accomplishment....which pushes them to move on to the next note etc...Other more technical skills are less easy to define as having "nailed" than a note.