Why amps, pre-amps, integrated amps???

A receiver is a tuner (to get radio), a preamp (selects what input you want to listen to and controls volume) and an amp that drives the speakers. Basically 3 boxes in one box. If the receiver has a phono section and a headphone section then make that 5 boxes in one box.
An integrated is a preamp and an amp in one box, with phono and headphone depending on the model.
Preamps simply select the input and control volume. And some preamps include phono and / or headphone as well.
Amps just take the input from a preamp and amplify, drive the speakers.
Receivers, preamps, amps, headphone amps, phono preamps all exist because "different strokes for different folks".
Most people on this forum would say that the best sound would come from 5 separate boxes (if you were interested in all 5 functions). I run an integrated amp with a separate phono amp. I'm not interested in radio or headphones. I might get better sound from a separate preamp and amp, but it would also cost a lot more, I'd need another AC outlet, more shelf space, etc. so practical considerations also exist on why people choose what they have.
There are slight variations (example an amp can take a signal directly from a CD player that has a volume control, things like that) but for the most part I think I covered it pretty well.
Oh yeah, and a CD player is also a transport and DAC, so you can consider that 2 boxes in one box also.

I was led to believe that better performance from separates could be achieved with being able to better split up and manage the power supplies. Not sure if that is true or not. In a general sense, it is always better to have more options so the more you separate components, the more options you have to mix and match and upgrade, etc. To me, I like the integrated approach although I prefer to have a stand alone phono stage that offers more flexibility than you will usually find in an integrated with a phono. An integrated also means you have less to worry about in terms of interconnects and power cords.

Each separate component has its own sound, and capabilities sonically, and mechanically (flexibility).

An integrated, or receiver, can be good for someone who likes the sound of everything ( each individual section)and flexibility it does.

I'll try to keep it simple with the three you mentioned:

The preamp has its own sound, certain types of needed inputs for everything you want to connect to it. A preamp is needed whether it's separate, or part of the integrated, or receiver.

The power amp section has its own sound, and output power limits to drive different types of speakers, or more than one pair if needed. It can be picked out to drive very demanding speakers an integrated, or receiver can't. In some cases, a power amp may not need a preamp, but it would need a volume control, and possibly a selector for multiple inputs like a preamp would have.

Every tuner has its own sound, and signal performance, and other features whether presets, or other preferred items.

Then you can get specialized. If you need a super good tuner for distant stations, or live in an area where the station you like has another one interfering with it. The separate tuner may out perform a receivers tuner, if this would be the case.

For the amplifier section. A separate amp can be picked out
to drive very demanding speakers, a receiver or integrated can't.

Then you could get more extreme (or specialized/custom) and mix tubes and solid state for sonic reasons, plus more variable capabilities and limits each type has.

You could use a strong tuner for a distant station, and low powered integrated, if you don't need the power for loud listening, or certain types of demanding speakers, for one example of using separates.

The list could almost be limitless for our personal preferences for all of each of individual component we choose limits for our preference.

So, if a receiver does everything to your satisfaction, you shouldn't need more. The same for an integrated if you don't need a tuner.

If you have a company known for great amps, another known for great pre amps, yet others building better tuners than others and on down the line then seperates allow you to go in whatever direction you wish. The reciever or integrated locks you in but folks with seperates can get the sound, features or even cosmetics in any combination they wish.

If your speakers need much power, you need a bigger amp(s) than an integrated can accomodate. It is difficult to find an integrated which equals the performance of separates. They necessarily involve compromises due to size constraints. The arguments in their favor are that they potentially allow one to eliminate a whole bunch of expensive cables, and usually take up less space. However, some "integrateds" have a separate power supply, which negates the size advantage.

You see an awful lot of "reason for sale - downsizing" ads on the Gon these days, as the boomers age and move into smaller places. That usually means getting an integrated to replace separates.