"What's considered "high output" please?"
From about 2.0 mV up.
"Why the heck would a company like Audio Note put a MM phono stage in a $4,000 integrated if there was no use for it!? Arrgh!"
There are lots of true audiophile MM cartridges, which many prefer to the MC sound. Some examples already mentioned. Others include Goldring, the Reson Reca, Audio Note's own retooled Goldring 1012 or 1022, as well as Grados priced at as much as $600 or so, with the Sonata possibly more than that. Shure and Stanton should be mentioned too.
High-output MCs are available from Benz (the Silver is a wonderful cart.), Sumiko, Denon (as somebody mentioned), Ortofon, and no doubt some I forget.
With a first-rate arm and turntable, you can get fantastic sound from a cart. like the Grado Silver or Gold, priced at around $150. In fact, the arm/table purchase can be worthwhile with cartridges priced considerably below those.