F=mA.
Apply some pitch from the cyclic via the swash plate. The increased AoA produces lift, a Force. This lift acts on the mass of the blade, and accelerates it upwards. But that acceleration takes time for the blade to change position. The maximum force applied, Lift, is around 90 degrees in advance of the maximum deflection of the blade. It took time, and the blade rotated around 90 degrees in that time. Lighter blades, like the R22, move a little quicker, and only need 78 degrees of advance angle. Each blade has its own forces and movements, and (in multi-blade systems) acts almost independently of its mates. The "disk" is the visual blur to our eyes, it isn't a real disk or a gyroscope.
A gyroscope is a rigid, solid thing, no hinges, no flexing, nothing that moves, other than a heavy mass spinning on an axis. A rotor system has feathering hinges, offset flapping hinges, drag dampers, the blades flex and twist, and all sorts of reasons why it only LOOKS like a gyroscope. Gurgle up Nick Lappos and read his "Helicopter Urban Myths", he has a bit more knowledge than most of us on Proon combined. Read John Dixson too.