The stiffer the power supply, the lower the headroom figures.
Depends on the current delivery capacity, regulation, amplifier circuit (internal resistance) as well as over-current protection.
That may be logical to you but generalizing such a statement will bring resistance to such generalizations.
I am quite sure you understand amp topology, power supplies, loads etc.
A high resistance in the source is not good, in the load it is far better.
I think you are often misunderstood because of the short, or lacking in depth, explanations and generalization of this topic.
One can build a very stiff power supply (regulated) and as long as the max. current capability is not exceeded, the amp circuit has a very low internal resistance near max. voltage output swing there will be small headroom numbers.
One can also massively over dimension a non regulated power supply and connect it to the same amplifier and get the same result.
Design choices often made for financial reasons (profit and/or budget range of the intended consumer group).
In this case the amplifier is what determines the sound quality (near clipping) and not so much the power supply.
The moment some form of current limiting kicks in (regulated power supply, non regulated sagging or amp limiting) you will see increased headroom numbers and non doubling output power at halving load impedance.
At that point one has reached the limits of the amplifier (as a whole).
From that point on sound quality (how the device handles that, i.e. what the waveform looks like) will differ substantially depending on design choices.
When this point is reached regularly one should simply buy a more powerful amplifier. It is that simple.
When you clip your amps often buy something else.
When you never reach that point there is no need for that and the stiffness of the power supply and max continuous power is all irrelevant.