How important is the pre-amp?

OP,
I had to come back to this thread. I just changed the preamp out of my system for a new to me unit and left everything else the same. Transformative is a massive understatement. This is one of the benefits of separates: The ability to pick one component and go for the home run...upgrading other things as time and budget allows. I may consider an integrated for the bedroom system simply because of space savings. I would sound like a total sound geek shill if I tried to explain all the differences this unit made in my system. My first experience with a true audiophile quality preamp has changed my mindset completely. Cheers.

So in the end, would a high quality integrated be the best way to go (assuming one likes the sound of the unit) due to one less pair of cables, electrical cord, etc?

Note the word 'best' in the question above. Integrated amps are a set of compromises. They might be excellent, but if we are talking about best then they will not be able to outperform separates that are built with the intention to be the best. This simple fact has been a thing in audio for the last 70 years and hasn't changed at all in that time. 

It would seem to me that attenuation would be best implemented remotely at the input of the amplification stage and all other control duties be placed further upstream.

In theory a well designed preamp should just act as a source switch and volume control. But does it add (or ruin) magic?

it also matches (and thus affects) source and output impedance to connected components and cabling, with sometimes significant sonic benefit (or cost)...

The preamplifier is at the heart of any great hi-fi rig.  A bit of history may be in order here.  In the 60's - the 80's nearly all preamps were full-featured. They included a phono stage, a line stage and a power supply all in one box. There are real advantages to this approach. The installation is simpler, cable requirements are simpler - it has a concentrated purpose.
All preamps of that era had single-ended circuit topology. That was the standard and had the very significant advantage of simplicity of the signal path. More recent marketing efforts promote balanced connectivity. Balanced topology imposes twice the circuitry into the signal path.  Abandoning single-ended is not an advancement.
Goals or attributes of a great preamp:
1) Straight line with gain without contributing to the sound thus preserving micro and macro details, etc.
2) Lo distortion
3) No noise (good signal to noise)
4) Two outputs
5) Adjustable gain and cartridge loading in the phono stage
6) Multiple stages of power supply filtering/regulation and a processor for control
7) Multiple line stage inputs
8) Line stage input with unity gain pass thru for Home Theater
9) Simple remote control
10) Able to drive long interconnects
This is not easy to accomplish by anyone's measure.  The market is full of preamps that fail pretty badly at either one or more of these goals.  After all, there are no standards and specifications are unlikely to give you any scene or what the preamp will sound like. Auditioning several will be best solution for choosing.
All of you in the "balanced only amplifier world" will have to seek out a "balanced" preamp.