What is the advantage of being able to select the MONO button?

I have assembled a dedicated mono rig (VPI Super Scoutmaster Special Edition w JMW Memorial tonearm, Ortofon Quintet mono cart into Plinius P-100 phono-pre with no mono button). I believe this to be described as a true mono cart and I’m getting different ideas as to what that means. Does this cart require a pre with mono button or is that just for playing mono records on a stereo rig?

According to the manufacturer:

Quintet Mono uses a strapped output to deliver the same output signal from both sets of pole pins. This effectively eliminates the need for mono-specific equipment, making it possible to enjoy true mono reproduction on any stereo playback system.

You don’t need a preamp with a "mono" button, it will play in mono through any stereo equipment and both speakers will output the same content.

What exactly does this button do if I had a preamp that had one?

The function of the "Mono" button in the preamp is to sum the left and right stereo signals into a mono signal for the speaker, so both speakers will output the same content.

What is all this about Y adapters? Do I need anything like this to optimize this set up?

No, I do not recommend using a Y adapter for the preamp output, it shorts the two outputs together and may damage the preamp.

The mono button does what the Y connector does to sum left and right.  You can then use the amp to feed identical summed signal to the two speakers of a stereo system or use one channel to feed a single speaker.  But, as I opined above, summing with a Y connector or a mono button is inferior to using a transformer to sum to mono.  You can google about the specifics of using a transformer for doing mono.  
 

I don’t have/want a mono system but I’ve heard many that are quite good.  Some mono systems were installed for practical reasons—e.g., a kitchen system where a listener can be in many different locations and still hear all of the music.  

So, if I understand this correctly, a mono button, y adapter, and "strapped" construction all do the very same thing: sum both sides of the cartridge together so that the same signal comes from both speakers. Correct? But why would a mono record not have the same signal on both sides to begin with? It's not made like a stereo record with a different signal on each side. 

Is the Quintet a "true mono" device? What exactly does this term refer to?

Is this cartridge safe to use on recent releases or just old records? Someone advised that the stylus shape is not correct for new releases and could prematurely wear them out.

So, if I understand this correctly, a mono button, y adapter, and "strapped" construction all do the very same thing: sum both sides of the cartridge together so that the same signal comes from both speakers. Correct?

Yes, the reverse-connected Y adapter and mono button combine the left and right channels, but the internally "strapped" construction cartridge is another story.

But why would a mono record not have the same signal on both sides to begin with? It's not made like a stereo record with a different signal on each side.

Most modern record pressing plants no longer have separate machines for cutting mono and stereo records, so most mono records produced after the mid-1960s have the left and right channels cut with identical content on a stereo cutter.

Is the Quintet a "true mono" device? What exactly does this term refer to?

According to Ortofon, the Quintet Mono is a "true mono" cartridge that generates voltage through horizontal movement only, as opposed to stereo cartridges which produce voltage through both vertical and horizontal movement. It "strapped" the left and right output pins together with a horizontal movement coil.

Is this cartridge safe to use on recent releases or just old records? Someone advised that the stylus shape is not correct for new releases and could prematurely wear them out.

The Ortofon Quintet Mono cartridge is fitted with a nude elliptical r/R 8/18μm stylus, which to my knowledge should be safe for use on recent mono releases.

I still do not understand why combining "strapping" the signal in the cartridge is any different than combining them in the phono-pre or combining them externally with a Y adapter. From the descriptions, it seems they are all doing the same thing but at different points of the signal route. And why would any of these methodologies be required with a mono record and cartridge? It seems to me that the signal would be mono anyway. Why do we need to combine identical signals?

If modern mono records have the same signal on both sides can't you simply play them on a stereo cartridge with the end result being mono sound? Is there any reason for one to use a mono cartridge on a modern mono record? Am I wasting my time and money pursuing this?