The modern mono cartridge feeds the signal, (vertical vibrations totally ignored), to both channels, so it goes to a standard stereo phono input and your existing stereo amp. You get full room sound, dual mono, frequency balanced as you have already solved.
So, using the mono amp might be fun, but not needed, and send the cartridge dual mono output where, how to a single mono amp, then to a single speaker? Unless you get a vintage single channel cartridge.
Not being critical of your idea, just letting you know my experience.
I have a pair of tube mono blocks (and spare), but far from integrated, they were made in 1958, so power switching, input switching, speaker out/in, all had to be independently solved. It was fun to have these 35 wpc tubes beat my 305 wpc McIntosh SS EVERY TIME! I sold the SS Preamp and Amp and got tube tuner/preamp from 1962 into modern integrated tube amp.
Another advantage of not using a stereo cartridge to play mono lps is that the grooves are different, and Mono cartridges are conical or elliptical, modern microline stylus are not 'right' for those grooves.
So, using the mono amp might be fun, but not needed, and send the cartridge dual mono output where, how to a single mono amp, then to a single speaker? Unless you get a vintage single channel cartridge.
Not being critical of your idea, just letting you know my experience.
I have a pair of tube mono blocks (and spare), but far from integrated, they were made in 1958, so power switching, input switching, speaker out/in, all had to be independently solved. It was fun to have these 35 wpc tubes beat my 305 wpc McIntosh SS EVERY TIME! I sold the SS Preamp and Amp and got tube tuner/preamp from 1962 into modern integrated tube amp.
Another advantage of not using a stereo cartridge to play mono lps is that the grooves are different, and Mono cartridges are conical or elliptical, modern microline stylus are not 'right' for those grooves.