Oh, Robert, what a can of delightful worms you have opened!
Speaking for my most ignorant self, I have always played —poorly, of course— with a flat-ish wrist, and know of no other way that works for me. If I have, say, 1% control of what I am doing this way, I have sub-zero control if I hover further above the strings. Obviously, this is no judicial statement on the Right Way of Playing.
Carlo (and the better players), however, manage to control the motion, keeping it down to a calculated, misleadingly easy-looking to-and-fro, "parallel" to the soundboard. I (and many folkie players I know) do ALSO use some "wiggle", especially in our "ethnic" tremolo. If I were to define and describe "wiggle" in terms everyone would readily understand, I mean the motion of a pianist's hand, playing octaves tremolo, thumb-to-pinky. I trust the image is unmistakeable.
I have never studied —or dared to, or found warranted, or deserved to, or ever thought of— with Carlo. The efficiency of his playing is astounding, and his artistry truly unique. As for me, however... I am doomed to wiggle for life. As a result, naturally, my wrist is LESS flat than Carlo's, giving me the requisite wiggle-ROOM; otherwise, I am, I suppose, of the "school" you describe, if only by default.
Needless to say, your mileage may vary...
It is not man who lives, but his work. (Ioannis Kapodistrias)