I don't understand the strong objection(s) to the primary/supporting method.

I think that if the primary/supporting system is taught properly, it's really
not too much different from the control/performance system that I prefer.
Note that the FAA's Instrument Flying Handbook states that the attitude
indicator (AI):

"... should always be used, when available, in establishing AND MAINTAINING
[my emphasis] pitch and bank attitudes".

However, it's been my experience, first as an instrument student then as a
CFII, that learning the primary/supporting method often leads to jerky,
overcontrolled flight. The problem is that the word "primary" is misleading.
Students tend to overemphasize the "primary" instrument and don't catch trends
early enough. For example, in straight and level flight, the altimeter is
"primary" for altitude, but it's really the least important instrument for
precise pitch control. A small pitch deviation, due to turbulence or
inadvertent control input, shows up first on the attitude indicator, then on
the VSI, and lastly on the altimeter. If a student concentrates too much on
the altimeter, by the time he sees a need for correction the plane can already
be in a pretty major climb or descent. This is similar to what happens to a
primary student who does steep turns staring at the altimeter instead of
looking outside at the position of the nose on the horizon. It's very easy to
get into a mode where he's chasing the altimeter, pulling the nose way up and
down, instead of catching deviations early using the outside pitch reference
and the VSI.

This is also similar to the problems a lot of students have while tracking the
localizer. They stare at the CDI needle and try to center it by reacting to
its movement, and end up chasing it back and forth.

As yet another example, although the airspeed indicator is "primary" for pitch
in a full-power climb, trying to use it for pitch control often leads to
oscillations as the student chases the lagging indication. This is
something I see a lot with VFR pilots on climb out, or in a simulated
engine-out glide. They try to chase the airspeed and are always a couple of
seconds behind, pulling the nose up and pushing it down instead of just
setting a pitch attitude, trimming away the control pressure, and then
checking the airspeed to see if the pitch needs a small correction.

Barry