I don't ever remember there being any issue with playing mono records on stereo phonographs, but apparently there was at one time a recommendation NOT to do the reverse. In other words, NOT to play a stereo record on a mono record player because it would ruin the record, or something like that. But I am not old enough to remember those days, so I don't really know this information for sure, firsthand, myself.
They said the same with Quadradiscs. But you get a totally different mix on Quadradiscs vs. Stereo LP's. Tony Orlando and Dawn's He Don't Love You album is a perfect example.
I don't have a CD-4 demodulator but play the CD-4 records off the turntable and they sound louder and better than the CD. It's too bad they stopped making CD-4 Quad.