Some flight trainers say that a pilot should separate the inputs on both axes, doing only one-dimensional inputs, i.e. pitch or roll inputs, but not at the same time. One possible reason for separating both axes might be a higher control accuracy when manipulation one axis only.
Well, first I would suggest those instructors go rent a Cessna and re-learn how to fly because that is the most useless statement on training I have ever heard. I agree with PM's sentiment of "muppet trainers."
Hell, the secondary effects of controls is what, Lesson 2 or 3 of a PPL...You really cannot get any more basic.
In fact, to answer your question of "some reasons...against the separation of axes" my answer would be the secondary effects of controls. Plain and simple as that. To do anything else is to forget and ignore the very basic principles of flight.
My god, what a joke our industry is becoming if this is really how pilots are taught. Any pilot who actually flies or teaches like that deserves the bottom of the barrel pay they are no doubt getting.