What is the benefit to keep a constant output range for VCO when frequency changes between the minimum and maximum values, especially for a wide control range VCO?  Razavi emphasizes constant output range in VCO design in his book. But I don't really get it.

Hm.. A couple of things that I am aware of -

Synthesis PLL's operate with low loop bandwidths and some amplitude changes there should not affect loop gain and associated stability and phase margin an awful lot. Some, but 3-6db of change should not destroy the system.

You have two different families of PLL's and they are
LC-VCO PLL's for synthesis and mixer use in the RF world,
and
Ring Oscillator VCO PLL's for timing and clocking types of applications.

With ring oscillator sytems you want a constant amplitude output in the form of a saturated state output prior to transitioning, otherwise you get a very noise (jitter prone) sensitive system.  Pretty much all ring oscillators are run this way.

With LC VCO systems you are back in sinusoid land and you are trying to maintain a spectrally pure sinusoid with minimal phase noise (and the associated spectral spreading) - That means to a certain point you are dealing with linearity issues (not too big in amplitude or you get harmonics) and signal to phase noise issues. (not too small in amplitude or you get a poor SNR and the associated spectral spreading)

All of the above scenarios point to a fixed amplitude being a bit more viable than variation in the signal output.

Probably some other reasons, but thats what comes quickly to mind.

Jerry