The most productive way to improve the headroom of an audio system is to add a good subwoofer. Got one?
But is it fair to leave aside power supply issues?
Receivers generally have just one power supply, and all things considered, this is a good thing. While amplifying music, it is unlikely that the number of channels being driven or how they are driven significantly affects the power available to the remaining channels. All of the numbers you've seen that show the receiver's output sagging when all of the channels are driven are based on non-musical waveforms and resistive loads that put an unrealistic load on the power supply.
You can check this out by popping the covers on your receiver and use a voltmeter to measure the voltage across its power supply capacitors. If the DC voltage drops by 10% or more, then that is somewhat significant. Ever done that?
What most commonly causes a receiver to clip/distort?
Amplifiers whether in receivers or not, are usually pushed into distortion by the gain being turned up to the point where the output stages run out of power supply voltage.
Is it the individual power amplifier stage asking for a higher voltage than its supply voltage due to preamplifier gain being too high?
Yes, that is the usual case.
Or is it the common power supply letting its output voltage dip due to excessive current draw?
Depends on how the receiver is being used.
If we are on the test bench driving resistors with a steady sine waves near maximum rated output applied to every channel, then the power supply will likely sag under the load and the maximum output of the receiver's power amps will be reduced on the order of 50%. This looks worse on paper than it sounds if you were crazy enough to sit around and listen to sine waves. 50% of the power is only a 3 dB loss which is just about audible.
If we are in the listening room listening to music through speakers, then the situation is vastly different. Music has at least 6 times less
energy than a pure sine wave with the same voltage. The speaker draws something like half as much power as an equivalent resistive load because the impedance of the speaker varies strongly across the audio band. Furthermore, multichannel recordings that drive all channels to full output and just sit there and grind, are very rare if they exist at all.
I'd think that reducing the demands on the common power supply could provide benefit to the remaining channels driven by the receiver.
That is true, if there is actually a problem.
And another concern I had with a setup like this is phase matching. Every time you run a signal through a power amplifier you get some kind of phase shift, right?
Yes, but that phase shift may be reasonable in the normal audio range, and large only at the extreme frequencies.
In the standard use case of a budget receiver like the one mentioned every channel is fed through an identical power amplifier, so their phase shifts should all match.
The common power supply neither helps nor hurts. The phase shift in the power amps is based on their own properties. One of the functions of a good power amp is to ignore what its power supply is doing, within reason.
But the addition of external amplification for only some of the channels would mean that some channels will have gone through one stage of power amplification and other channels will have gone through two stages. Is this cause for concern?
Generally no. Power amps just don't have that much phase shift in the normal audio range, and the phase shifts due to the speakers and different locations in the listening room are always far larger.,
If you are sitting at different distances (inches, feet) from your speakers, the sound from them undergoes massive shifting due to the time it takes sound to pass through the air. 1 foot of air causes about 1 millisecond of delay, which causes 360 degrees phase shift at a relatively low 1 KHz. At 10 KHz its 3,600 degrees! At 1 KHz the phase shift in a power amp is just a few degrees.
Many 2-way speakers run the tweeters with reversed polarity as compared to the woofer. The phase shifts in speakers and rooms are massive and tend to wash out whatever little changes happen in the power amps.