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

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?

You don’t “need” a mono button to play mono records.  Whether or not playing a mono record using information from the sum of two identical channels vs. just one channel is a different matter (I have no idea).  You need to sum two stereo channels to play stereo records in a mono set up.  Not everyone who has a mono setup only listens to mono recordings in such systems.

A true mono cartridge has on one signal generating element that responds only to side to side movement of the stylus and ignores vertical movement (which is information in stereo but just noise in mono).  Modern mono cartridges have suspensions that allow for vertical movement of the cantilever even if such movement does not generate a signal because this acts like springs and shocks on a car—it allows the cartridge to better track vertical movements from warps and other bumps and reduces stress on the stylus and cantilever from such movement.

You do not need a mono cartridge or mono switch . Mono reissues are cut on stereo cutter heads in 99% of cases so you don’t need a mono cartridge for them. They’ll play fine with a stereo cart. If you have original mono pressings recorded 60 years ago that is a different story!