Elephants and humans can both get the same strains of TB.
Mammoth remains show that many had TB.
It was shown by one study that Asians crossing the bering land bridge probably carried TB to the mammoths in this hemisphere.
More likely we can blame the ignoble microbe than the noble savage.
BTW: why didn't humans in Asia and Africa kill off all the Elephants? Same kind of problem vis a vis smallpox and Indians. The elephants had evolved right along with TB and adapted. Over here, the mammoths never had encountered it before.
But there were other causes, for extinction, too. We were coming out of an ice age and many of the creatures such as the giant beaver, simply couldn't adapt.

I, personally, get quite ill with every "Human bad, Gaia good!" theory that comes along. Pardon me while I puke.