It has always been a combination of the two, I see the point for teaching students, it is simple to teach stick for speed and throttle for ROD.
But if you take level flight, throttle = speed
And I was taught on my first lesson that if you push forward "cows get bigger"
Most of us without thinking, reduce power to a setting that when we push forward to a 500 fpm ROD, we maintain a certin speed.
If we were to hold 500 fpm ROD and increase power then we would speed up, and if we reduce power and hold 500 fpm ROD we would slow down (unless we had a best glide ROD of less than 500 fpm).
I think we all actually use throttle for speed and stick for ROD but it is easier to trim for 80 knots than to trim for 500 fpm ROD.
Confused??
Tony