The way you do it and the way Brett did it are largely the same thing, the difference just lies in where you decided to break up the numbers and when you added the numbers you moved around back in. You went to 900 instead of 500. They’re both essentially the associative property in action, though neither of you actually show that in your illustrations of your thinking.
Technically, Brett’s probably should have been illustrated as something like “489 + 916 = (489+1) + (916–1) = 490 + 915” and “489 + 916 = (489+11) + (916–11) = 500+ 905” to more explicitly illustrate the changes.
That’s where the number sense comes in, though. It’s the knowledge that you can do either of these methods and come up with the same answer.
As for who learns how, one thing I can attest to is the fact that “the old way” didn’t work very well for me. I could do it, but timed tests? Failed every single time, solely because I couldn’t come up with the answer fast enough, and rote memorization was somehow supposed to fix that (because every time you fail it, you had to write the uncompleted and incorrect ones a few dozen times). The problem was that for me, it was never about not knowing that, say, 8*7=56, but that it took me half a second to dig that out of my mental memory bank, and when you have 60 seconds to answer as many problems, half a second in mental processing is about a quarter of a second too slow.
Personally, I did much better once I got into Algebra level math, precisely because “number sense” was taught and my teacher strove to teach the “whys” and “hows” of the various algorithms and formulas that were used, and shared with us the ways that he had learned over the years to solve particular problems. This understanding is what served me best when I became a math tutor/faculty assistant in college and was teaching these things to adults going back to college, for whom “the old way” again failed them, as they had forgotten this level math, or never understood it to begin with, and needed remedial classes.