This kind of approach is familiar to me from competition math. Back in middle school math club we were taught this exact approach (under the name of "Vieta's formulas"), i.e. that thinking about the sum and product of the roots could be faster in some cases. Po-Shen Loh is the director of the US IMO team, so it makes sense he would like this approach.

However, I don't think it makes logical sense to teach it only this way. Here you start by assuming that a quadratic has two roots, which is not at all obvious the first time a kid sees a quadratic equation. (Especially because those roots can be complex numbers!) Completing the square tells you why there are two roots, and also naturally leads you to the necessity of complex numbers, i.e. when the "square" you end up making is negative. You can use the nice Vieta's formula tricks only after establishing that.