A lot of what you describe has become the standard way to teach elementary math-- from the "new math" onwards to Common Core pushing aspects of looking at problems the same way
Our school does the much-lauded Singapore Math in elementary, which definitely tries to build intuition and looks at many approaches, and supplements with drills.
And I teach a competitive math class which definitely is all about finding different ways around problems and comparing and contrasting.
I'm thrilled to hear it's popular! In my hometown it was killed by doubters, but perhaps with expanding evidence they'll reconsider. I think perhaps it was lumped in with disastrous testing efforts, but the math at least was pretty great.
Our school does the much-lauded Singapore Math in elementary, which definitely tries to build intuition and looks at many approaches, and supplements with drills.
And I teach a competitive math class which definitely is all about finding different ways around problems and comparing and contrasting.