It does not matter much how far they are from grenadiers (probably BCLG is researching that as we speak, since he got gunpowder a few turns ago), or military tradition for that matter either.
Both these units (though the grenadier still has a tough time vs cavs) are simply too expensive for their economy to produce en masse, so even if they have the proper technology, my assumption is they simply will not be able to field enough of them to get out of the hole they are in now.
And even if they do (mass rushing/upgrading) it will set them back even more in the long run, as it will cripple their future output/research while I set course for flight. Might be I'm too optimistic here (which usually really isn't my style) but those graphs and F9 screens speak for themselves...
Take for example Darkness, who atm has 1 size 11 city (which completed the Haga Sophia a few turns ago
), 2 size 7 cities, 1 size 4 city and a size 2 city, of which 2 are currently besieged and in danger to fall. That leaves one size 11 city and 2 size 7 cities as his productive core + research capabilities. With those 3 cities, he needs to fend off my raiding parties, the American onslaught ánd create an offensive army to besiege American cities. In between, he would also somehow magically need to create grenadiers out of thin air... it simply seems impossible to me.
BCLG is currently in slightly better shape, Cuzco is a strong research center with a lot of villages/town around, though that results in slow production there. He also has 3 other decent cities (size 7, Corihua, Elephantine, Thebes, plus 2 size 4 cities (Memphis, Machu Picchu). He also seems responsible for the research at the other continent now, and I expect him to join in in fighting America soon, considering his severe power-buildup of the last few turns, which as said drains his economy. Maybe he intends to capture american cities then gift them to Darkness, since I don't think he can afford to sustain more cities with his current empire spread.