It all depends how careful you are
Dustdevil
The problem comes if you are carrying the velocities to the next time step in a moving axis system. If you do that then you are violating a number of laws of physics.
Example:
Aircraft in level unaccelerated flight, angle of attack zero (for simplicity).
Introduce a perturbation in vertical velocity (down). This causes a small increase in alpha, which for a stable aircraft causes a nose down pitching moment. The aircraft pitches down, so reducing the angle of attack because the velocity stays pointed slightly below the horizontal and now the nose is also below the horizon.. Depending on the aerodynamics/inertia you'll then get SPPO and Phugoid motion in some combination - just like you should.
Now keep track of velocity in aircraft axes, not earth azes and try again. It's all the same until...
aircraft pitches nose down. As the aircraft reference rotates so do the velocities in that axis system and the angle of attack does not reduce. Aircraft happily bunts over into the ground!
If you are handling all the transformations between axes correctly then it's ok to use whichever are convenient for the item concerned. But you have to have a set of inertial axes somewhere.