I think you may be conflating rote memorization of math facts with the algorithms used to calculate solutions. Teaching one does not require excluding the other.
The initial example in the article was likely used for its simplicity (so as to not scare off readers!) and its daily applicability. This mental math relies even more heavily on rote memorization than algorithmic teaching does. The person doing the calculation has to be able to manipulate the numbers meaningfully, and this requires a solid foundation of rote memorization.
To me, the real value of common core math is its reliance on logic. The Gauss formula demonstrates the value of a logic-based approach; the students who hear this story will remember the logic behind the formula and should never struggle to recall it later in life. The math is actually identical, but the teaching approach is radically different.
The benefits of logic extend well beyond math classrooms. We benefit from thinking more about Why and less about What. It obviously helps to learn computer programming, which, at its foundation, is nothing but logic statements. However, there are countless applications which are less obvious. Logic helps students develop a bias toward trying to understanding the root cause of a problem so that they may work toward a solution. This skill is relevant throughout life, whether a student aspires to be a diagnostician, an environmentalist, a politician, a military leader, a pastor, or any other field you can imagine.