I have always understood a power supply will maintain its output voltage up until it reaches the output current capacity.
I really think your missing something. Two basic power supply topologies are CV, CV/CC and CV/CL where C?=constant; L=Limit and; ?C=Current.
In a Limiting situation, it's not regulated and CV could just be protected by a fuse. The Power supplies will have an SOA or Safe Operating Area.
There is a mechanism known as foldback current limiting: tps://en.wikipedia.org/wiki/Foldback_(power_supply_design)
You may find SOA or Safe Operating Area curves for power supplies. They may be in various forms. I-V and I-T are possibilites.
https://www.tij.co.jp/jp/lit/an/slva766/slva766.pdf The SOA can be clipped at the corners too.
There are "power supplies" names "Voltage sources" and "current sources" or SMU's that operate in 4 quadrants. e.g. I+V._, I-V-, I-V+ and V+I-.
The HP DC power supply handbook
https://literature.cdn.keysight.com/litweb/pdf/5952-4020.pdf is a really old reference, bit a very goos one.
I've read about a new design and just can;t place who with some very strange SOA curves.
Nothing is ever simple.
There are times that 3 resistors in series are not the same as a single resistor or times that mounting a SMT component edgewise is a different component because of parasitics.
Wires that wiggle in the earth's magnetic field generate a current. Moving wires generate a current. Most people don;t have to worry about those effects- I did.
There are circuits that can be altered if you touch the components.