Yes, that's right, the comparator controls the amplitude. However, at higher frequencies, the amplitude does tend to increase, due to the propagation delay of the comparator, the output tends to overshoot somewhat.
There is a variation where the output from the integrator drives the other comparator input through a resistor so bypassing the resistor with just the right capacitance triggers the comparator earlier as the frequency rises compensating for the delay and linearize the frequency versus voltage.