Ive found those with substantial often times, over built perhaps, power supplies do indeed contribute to the stability and performance level of an amp . Its just NOT the only thing which does however.
The power supply becomes still more important if the speaker mated to it is not optimally chosen. The PS is responsible for supplying sufficient voltage/current at whatever frequency immediately to the speakers so they will reproduce an undistorted signal. Differing freqs need more power to generate the same volume than do others. Consequently, the power sup must be able to provide the required juice or problems develop, like clipping or distortion. Neither is good and either can be damaging . Especially to the speakers.
A fair enough rule of thumb for selecting class A or A/B amps is just to see how heavy the amp is. Again, its not the only aspect worthy of consideration but its right up there in those type amps. Capacitors too play a big part, size, type, and number of them add to the power sups ready & reserve energy supply.
Hope that helps.