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

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!

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.

Unlike adapting a stereo cartridge for mono either through rotation of coils, or summing of channels (perhaps to reduce manufacturing cost?) and sold as a mono cartridge, a true mono cartridge only using a single coil (2 leads) and "strapped" to two sets of output pins (so ​​it can be connected to a typical tonearm with 4 leads).

Simply put,

Some so-called mono cartridge strapping two channel together internally in order to output mono signal.

A true mono cartridge strapping mono signal output into two identical channels.

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?

Yes, but there’s a drawback.

Is there any reason for one to use a mono cartridge on a modern mono record?

The biggest benefit of a dedicated true mono cart is that it doesn’t pick up noise from vertical motion, which a stereo cart can’t eliminate.

Am I wasting my time and money pursuing this?

It depends on how many mono records you own, how often you play them, and whether you can tolerate a little more noise.

If you are starting with a mono recording and a mono cartridge, use will only need one channel of the preamp and a mono amp with one speaker. But if you are playing through a stereo amp and two speakers it gets a little more complicated.  For that setup I would put a Y connector on your turntable output to run both channels of the preamp and stereo amp.

The Mono switch is not needed because you are only sending mono from the cartridge.