The simple answer is that the predominate force in rotor dynamics is aerodynamic and it is easier to explain using aerodynamic principles and end up with a more correct layman's explanation than the opposite. ie. You start flying the blade up at the tail to produce a left roll.
The more complex equations account for conservation of angular momentum because they have to, but gyroscopic precession is an over simplification of the effects of conservation of angular momentum because with the exception of a teetering rotor head, the system does not act as a rigid body which is an essential assumption of gyro behavior.
Further:
One thing that comes to mind is running out of control authority in fast turns. How does that get explained in the dynamical blade picture? It seems more natural to describe that again using collective properties of the disk.
I'm assuming you're talking about lack of left roll control authority in a high G right turn (assuming a counter clockwise rotating rotor system) which is ironically the exact situation I was going to bring up to counter your gyro discussion. How would you explain this in a rotational dynamics concept? I can explain this quite well within an aero forces construct. Simply put, as I increase the G on the head, coning increases. As a result of increased coning the aft portion of the disc sees an increase in induced flow/drag and a subsequent loss of lift while the front portion of the disc sees a decrease in induced flow/drag and an increase in lift. The result is the blades passing through the decreased lift region (aft) will tend to flap down and those passing through the increased lift region (front) will tend to flap up. To counter this pilot would have to apply increased left cyclic. As a result a sustained high G right turn in an American helicopter will tend to cause the cyclic to "migrate" left as G increases thanks to increasing coning, which will of course eat up total amount of left cyclic available and hence a reduction in control authority. The key to "fixing" this is to reduce the coning to restore control authority which can be done by either reducing collective or unloading the disc by adding forward cyclic or a small contribution of both.